Any good cheap integrated amps?

Well, good tuners are rare in low-cost receivers. Tuners in general are very location-sensitive, however, and you could well find one that would work for you. Take a few home on trial and see which one pulls in your favorite stations best.

For what it's worth, several years ago The Audio Critic conducted an ABX comparison of a 100w/ch Pioneer receiver (gasp!) and a $2500 power amp. Small, informal panel, but nobody could tell the difference--even though, sighted, they were sure they could tell the two apart. Now I'm not actually endorsing Pioneer receivers here, but the successor to that unit goes for well under your budget, and may serve your purposes very well.