I am in or have recently graduated from a PhD program. Do you have any advice for me?

[Here, I'm not going to try to convince you that you want a job in this field. If you want that, though, here's some advice on how to get one.]
For better or for worse, people have weird biases around PhD candidates/graduates. Some folks think that it’s really great to have the additional academic training and a solid grounding in substantive theory. Other people will be worried that you can’t respond to the fast pace of a campaign and are too theoretical to be useful. And, if we're being totally honest, a fair amount will be wondering in the back of their heads whether you will see your skills and time in graduate school as more valuable than folks who've spent years on campaigns. If you can, try to find out which camp your organization or interviewer falls into. One really easy way to do this is to see how many PhDs (or PhD dropouts) they employ and if there are any from your field. Reach out to them as a fellow PhD holder: you'll probably get a fair amount of sympathy for purusing a non-academic path.
No matter what, try to frame your research and work in practical, not theoretical, terms. What is the value of the knowledge that you bring for campaigns? It’s okay to say that there isn’t a lot of practical value, but, in that case, what is the value of the skills that you bring? -- The point here is that you need to demonstrate that you will add value to an organization, not just do research that people won't find useful.
As a PhD candidate, you’ll be held to higher standards than other entry-level candidates, even if this is your first job out of school. You should be comfortable with a variety of statistical methods and know either Python or R. If you only know Stata, then take 2-3 months and learn another language. List that language first on your resume. Graduate school is a near-optimal environment to learn hard skills, and you should take advantage of that.
In your cover letter, be explicit about what you’re looking for. Is it a job for the election cycle? A summer + fall semester job? Looking to leave academia?
A lot of your colleagues will be younger and less credentialed than you. There is a reason they have the job they do, and you should learn as much as you can from them (and they should also learn from you). Don't be condescending and don't assume that that, by virtue of your training or credential, people ought to defer expertise on politics to you. If this would bother you, then you might want to stay in academia.
Some misc and very specific advice:
  • Don't use jargon and don't assume that people know what "endogeneity" (or similar) means. Most interviewers won't ask you to clarify (because it's hard to admit that you don't know something!), and will just assume that you're a bad communicator.
  • Relatedly, if asked for a writing sample, be wary of just sending a dissertation chapter. If you're asked for a writing sample, it's probably because the job requires communicating with non-technical readers. Is a dissertation chapter really a good example of that?
  • Try to pump yourself up before your interview. I know the PhD probably wasn't a ton of fun (hence why you're trying to get out?), but try to seem excited for this new opportunity.

Should I go back to school (to develop technical skills)?

According to the 2016 salary survey, people with PhDs tend to make higher salaries in this field, but there's not a big difference between Bachelor's and Master's degree holders (see Table 22). That said, it's hard to say whether there are causal relationships here.
For most jobs in this space, I would be surprised if it were necessary to go back to school. It may be easier for you to learn certain skills in an academic environment, but a lot of those skills can be self-taught or learned in an online course. But self-teaching is hard. You may also like school! If you have the money and want to go back to school, it's unlikely that going to school will significantly hurt your career outside of the opportunity cost of not working for X number of years (this is generally more true for PhD programs than Masters programs). One option, if you can't decide, is to audit courses at a local university.
There are some roles where formal academic training will be necessary, and I think this becomes more true the more specialized you want to be. If you want to develop algorithms full-time, for example, some formal training will probably help. But most "data science" problems that organizations have can be solved in Excel or through some fairly basic open-source packages in R or Python.
A note on online certifications specifically: I am sometimes skeptical when I see certifications from online courses listed on resumes because, typically, the theoretical training is weaker than in an academic institution, and the practical components are as weak. I am most impressed when people take statistics or programming skills and apply them either to their work or to non-work projects involving real data. That's just because working on an unstructured project with messy data is substantively different than completing class assignments. In lieu of listing online courses, I would just list the languages/programs that you know at the bottom of your resume and describe, in the body of the resume, some projects you have done using those skills (within or outside of past jobs).

How do I reach out to someone?

The progressive analytics community is generally pretty nice, and you should befriend other people in your field - senior, junior, etc. They may not be your mentor (or even your friend), but everyone likes talking about themselves and you shouldn't be afraid to ask someone to get a coffee/drink to just chat.
That said, you want to mindful of the fact that most people are busy, and their time is valuable. Don’t reach out for the sake of reaching out. Is there a particular reason you want to reach out to that person? Do they have some experience that particularly interests you? Do your homework and have specific questions in mind. In general, you should plan on no more than a 20-30 minute conversation, and you should show up/call in on time.
In terms of contacting people, you will always be more successful if you go through someone you mutually know or if you can point out a common connection (e.g., college, high school, mutual interest, etc). But even if you don’t have those, you are at least mutually interested in progressive politics!
Here’s a sample email for the initial contact that is polite, respectful, and brief:
Dear Jordan,
My name is Jenny, and I am a [role at organization or university]. I saw that you [how you know of this person], and I am interested in [why you want to talk to this person/why this person should care about you]. I was wondering if I could ask you a few questions about [topic of interest]. My resume is attached here.
Do you have time for a short conversation the week of __? I am free [Monday 4-6 PM EST], [Tuesday 10 AM - 12 PM EST], or [Wednesday 5-7 PM EST]__.
For example,
Dear Jordan,
My name is Jenny, and I am a senior at Babayaga University. I saw that you’re working at EMILY’s List, which I first learned of when I was a volunteer for Barbara Mikulski’s campaign in 2010. I am interested in pursuing a career in progressive analytics, and I was wondering if I could ask you a few questions about how you transitioned from field organizing to targeting. My resume is attached here.
Do you have time for a short conversation the week of February 1st? I am free Monday 4-6 PM EST, Tuesday 9-11 AM EST, or Wednesday 5-7 PM EST.
If you don’t get a response to this email, it doesn’t mean that this person doesn’t like you. It’s probably more likely that they’re just busy. At any rate, it can’t really hurt and can possibly help.

What happened in 2016, and why would I still want to join this field?

The first question is solidly beyond the scope of this doc. The second is that, despite whatever problems you believed occured, it's clear that we have a lot of work to do in the coming years. Want to help?