How can I do solid, publishable research without having a postdoc position?
This is a good question but there may not be a good answer. Students who graduate from strong programs with multiple publications and strong letters of recommendation still have almost no choice about where they have to go to pursue a postdoc, and the same problem arises 2-3 years later for tenure-track positions (if they manage to get one). It does not sound like your family is OK with the amount of uncertainty that pursuing this path would require.
For example, even if you were the strongest student graduating from Harvard, to pursue the postdoc-then-tenure-track course you and your wife would need to be OK with (1) moving next year (2) to a location you don't get to choose, then (3) moving three years later (4) to a permanent location you don't get to choose. (If you're lucky, you might get to decide between a couple options, but none might be where you'd want to live.) Hopefully you get tenure there, otherwise you'd (5) have to move again after six or seven years.
My experience is with research positions, and the situation does become somewhat different as you move towards teaching-focused positions like small liberal arts colleges, or community colleges. But it still sounds like staying in your current position may be orders of magnitude better for you and your family than any other option.
One thing that stands out about your post is that although it's clearly and cogently written, it is vague about the key point of what you are looking for. Do you want a job where you teach stronger students? or more advanced classes? A job that affords you time for research? or which evaluates you on it? Or do you just want to do research and publish? I would suggest taking the time to think about what exactly your goals are; hopefully some of them can be achieved without changing positions.
As for publishing: the academic publishing industry in mathematics is not so hard to deal with. The difficult part is finding a problem to work on and proving a theorem! Once you do that, given that you have a PhD in math you should be able to look at papers on the arXiv to see the style in which math papers are written.
I strongly agree with everything in Tom Church's answer, and want to add to it a little bit.
The market for postdoctoral and tenure-track jobs is, I'm sorry to say, extraordinarily difficult. In your own words, you "limped and hobbled" your way to a Ph.D. Unfortunately, even people who sail over the bar often have difficulties finding academic jobs.
One piece of advice I would give to you is to attend conferences. There are plenty of conferences in a variety of mathematical disciplines, held all over the US and in many other places. These are almost always free to attend, assuming you can afford the travel costs. (And occasionally the conference organizers have grant funding to pay for those too.) At some, you can hear hour-long talks given by leading reseachers. Others have slots for 20-minute "contributed talks", and you can present your thesis research and listen to other talks given by junior researchers. Some conferences combine these formats.
These are advertised on the web, so a bunch of Googling ought to turn up some conferences you might be interested in.
The mechanics of publishing per se are actually simple. Once you've written a paper that is good enough to publish, you might e-mail a copy of your paper to someone whose work you cited and ask their advice on what venue to publish in. What is vastly more difficult is learning to write good papers. I second Tom Church's advice to browse the arXiv and read papers in your area of mathematical interest.
Finally, I hope this is not rude, but -- count your blessings. You say that you "currently work for an incredible employer and a great college". Not even all Ph.D.s who do find academic employment would say this about their employers. I would recommend looking for opportunities in your existing institution to expand the scope of what you teach. (Especially since, as you say, moving to a different city is not an option.) Perhaps get a sense of what there might be student demand for, and then speak with your chair or dean? You might not get to teach analysis or abstract algebra, but you might be able to expand the scope of what you're doing a little bit.
Good luck!
Congratulations on finishing!
"The only school nearby that has postdocs will not hire me." Putting this together with your partner's unwillingness to relocate, if you want to do original research, you'll have to do it outside a postdoc. That is not necessarily a roadblock.
You can work on your own with the help of a mentor, and you can collaborate. You can do both of these things remotely, in principle, with perhaps some visits.
Your advisor may be able to help you find a mentor.
Read papers you find interesting and start corresponding with authors.
Go to conferences and make connections.
On the topic of teaching higher level courses, you can apply to teach at other schools without a postdoc and without publications. There's no guarantee you'll be hired, but your chances are better if you:
have outstanding references and student evaluations from your current employer
are willing to work as an adjunct, at least at the beginning
can show a commitment to community outreach, as well as to diversity
are willing to teach any course the department throws at you
Here's a fun idea: sometimes there is a group of adventurous high school students who want a math club and need an outside advisor willing to work with them on more advanced math topics than what they might see in their high school classes. If you decide to work with such a group, it would probably be on a volunteer basis.