I've done the whole visa reseting on January 1st thing, and your days do indeed reset after the new year. If you play your cards right, you could potentially stay in Colombia for up to one year. A few years ago, before 90 day visas were the norm, I had to make a short trip to Ecuador because my 1 month visa was about to expire. I just stayed in Otovalo for a few days and crossed back into Colombia with no problems.
You can find work as a teacher without a visa. Many companies pay in cash and aren't worried about such things. However, if you want to stay for a protracted period of time, then you're eventually going to need a visa. I have seen advertisements for schools offering visa to foreigners who are willing to sign a contract. Check out the Colombian Craigslist education section. Many institutes post their openings on that page, and I frequently see companies that offer visa sponsorship. Do be careful though, I've heard stories of these companies taking advantage of the teachers once they are in Colombia.