I largely agree with what RoboKaren wrote (that the cover letter is key, and it and the CV should emphasize teaching over research), but I want to add a bit to address the OP's last paragraph, regarding research statements. I'm a math professor at a small liberal arts college (SLAC) in Ohio.
For a SLAC, there is a bar for teaching and a bar for research. You must be above both bars (and it helps to show you are gung-ho about service too). Different SLACs will have different bars. If you're above both bars, then it's very unlikely that having "too much" research would be held against you. I say this as someone who has published 30 papers in 10 years, when 10 papers would have already been "exceeding expectations." Good SLACs want strong research faculty. But be sure to make the case that you actually want to be at the SLAC you are applying for, so they know you're not just going to leave right away after taking the job. Let me turn to the last paragraph of the OP:
Also, I understand that it is inappropriate to give an overly detailed research statement. However, as a representation theorist, I think it is to my advantage that my research draws on a lot of different fields. Is it better to explain the broader relevance of my research, or do I ignore that in favor of what I can teach undergraduates to program on a computer? Does it make sense to write separate statements?
I recently wrote an answer about research statements for SLACs. I argue that it's good to start off very basic and work your way up to your research accomplishments. The hiring committee is probably using your research statement to decide if you're good at explaining things and would be a good teacher, as well as to decide if you have enough of a research trajectory to get tenure. So, yes, you should write separate statements. I would certainly explain the broader relevance of your research. Anything about teaching students to program can go in the teaching statement, but ideas for how to include students in your research program can be a section of the research statement.