First, let me say that I'm impressed by your maturity and wisdom. It's not easy to recognize your own limitations, accept them, and adapt. Most people have to learn the hard way, by living through a few decades of struggle and frustration. Some people actually enjoy struggle and frustration, though. Your choice.

I have been an "industrial" mathematician for 40 years (I "sold out" long ago). I work in the software industry. I don't have a Ph.D, and I don't write research papers (not very often, anyway). I don't spend a great deal of my time doing mathematics, and almost no time at all doing original mathematics, but I do write "mathematician" on my tax return every year. My work is interesting (to me), and I've made quite a lot of money.

From the suggestions below, it's clear that some people judge the fabric of a profession by reading its research literature. This is a hopelessly misleading approach. What happens in day-to-day work in any industry is very far removed from what you read in research papers. If you want to know what it's like working in industry, you should ask people who work in industry. And this is not a very good place to do that. Most of the people who hang around here are university faculty, grad students, and (recently) kids trying to get someone to do their homework for them. If you want to know what software developers do, for example, ask at StackOverflow.

I'll repeat some of the advice from others. Learn some computer science. Learn about basic algorithms, and get good at programming in some mainstream language like C/C++ or Java (not Haskell or OCaml). It's not that difficult, and it's great fun when your code works.

Accept that no-one is likely to pay you to do original research (except on a very small scale as part of a larger project). Especially not mathematical research. People in industry are expected to create working saleable products/processes/systems with a high degree of predictability. Research is too risky. If it were less risky, and the results were more certain, then it wouldn't be research.

Think about what it means to "sell out". One definition says that selling out is doing what society (and your employer) want you to do, rather than what you want to do. But society (or your employer) will only be willing to pay you if your work is valuable to them. So, in some sense, selling out is inevitable unless you're going to be a hermit poet or you're independently wealthy. The best you can hope for is that your work is interesting and fulfilling (in addition to being valuable), and that you don't have to do anything that you find morally distasteful. If you think that making money is distasteful, stick to academia.

To answer your question, I'm not personally aware of any places in industry where significant numbers of people spend time pondering the workings of abstract algebraic constructs. I don't say that they don't exist -- just that I'm not personally aware of them. My mathematical work mostly involves differential geometry (in 2D and 3D space), approximation of functions, numerical methods (root finding, minimization, etc.), very simple linear algebra, and occasionally a bit of algebraic geometry. I very rarely do any original mathematics. I typically use software packages written by other people, and I only need to know enough mathematics to understand the limitations of these packages and their applicability to my problems. If you want to work on the development of the mathematical software tools used by people like me, check out companies like Wolfram, MathWorks, MapleSoft, Rogue Wave, NAG. But be aware that these are (mostly) fairly small companies and they don't employ very many people. And they won't hire you unless you have good programming skills.

I mostly work with manufacturing companies -- people who design cars, airplanes, consumer electronics gadgets and so on. Think about what those companies are trying to do -- they want to create more attractive products, more quickly, with lower costs. How can you (and your expertise) help them do that? Contemplate this until you identify some place where you can imagine that you might fit in and be happy. Or, pick some other industry and go through the same sort of reasoning. The key is to find some place where your skills can add value.

Stop thinking of your work as your life. You'll still be the same person, regardless of whether you're winning Fields Medals or hacking code. Your children will love you just as much either way.