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.