engine idle problem Options

[RumpleForeskin05]

well ill start from the beginning so you can fully understand, I bought a 94 celica for relatively cheap. It didnt take long for problems to happen. From a cold start it was fine, but when the engine warmed up it would idle up and down and sometimes it would die. The problem got worse and it would die everytime I hit the clutch if i didnt rev it while clutching. A couple days after this started I got stuck in some snow and tried getting out, blew my engine even tho i never went past 5000 RPMs. I bought a JDM 3SFE and just completed the swap but the problem still persists. Whenever the engine is hot (not overheating, the temp guage is less than halfway towards the top) it will idle at 1100 for a bit, then down to 500 or below, then when it gets real low the car will shake badly and sometimes after that itll go back up or just die compltely. help me out PLEASE

[RumpleForeskin05]

ok thanks ill get all that checked out, starting with the O2 sensors, anyone else have anything to add?