Jason I checked on some rates -
Most of the Trail courses in January or February are $55 including cart, $40 walking (these courses are generally not very walkable). Ross Bridge in Birmingham and the Judge course in Prattville are more expensive. I'll bet there are Trail packages you can get that are cheaper than this though. This subject has been discussed on here before, but I like the Auburn and Greenville Trail sites the best. None of them are great GCA by any means, but they are a good value and fun to play, sort of Doak 5-ish, give or take.
For the other courses I mentioned earlier:
Limestone Springs GC (Oneonta AL, about 35 minutes north of Bham): $59 weekday ($25 replay), $69 weekend ($35 replay). Limestone and most of the others have cheaper twilight rates (which probably won't do you much good in Jan or Feb.).
Ballantrae GC (Pelham; 25 minutes south of BHM): $47 weekday, $64 weekend.
Timberline GC (Calera AL; 35 mins south of BHM): $52 M-Th, $57 F, $62 Sat-Sun. Very walkable.
FarmLinks GC (Sylacauga, 45 mins SE of BHM): $100, all the golf you can play plus lunch and range balls.
Ol Colony GC (Northport; right near Tuscaloosa, an hour west of BHM): $47 weekday, $53 weekend.
Capstone Club (Brookwood - 20 mins from Tuscaloosa, 40 mins from BHM): $39 weekday, $49 weekend.
One other I didn't mention before is Twin Bridges, a Gene Bates designed muni in Gadsden AL, #8 GD Best New Affordable in 2004. An hour NE of BHM. $33 weekday, $43 weekend. Very walkable. This course pairs nicely for a daily outing with Silver Lakes, the Trail course near Gadsden. You'd have to get a very early time at one course or the other, but I think you could fit both in on a day in January or Feb.
Round trip airfare from Minneapolis to BHM is probably going to be in the $400 range. I didn't check on hotels but there Hampton Inns, Courtyards etc. all over the place.
Average highs in January are in the 50s, low to mid 60s in Feb. Doesn't rain much that time of year.