Shuts off while driving?

I have a 05 Escape 4cyl, auto, 95,000. I bought it used at 28,000 mi. It runs fine when you start it, and it's cold. After you drive it for awile, and everything warms up, it tries to stall when you come to a stop sign. Half the time, it does stall. I have to put it in neutral, give it a little gas, then pop it into drive to get going. Then, it'll run fine until I stop again. Also, if I put it in park start it, it starts fine. Then switch to drive or reverse, it'll stall out. Only does any of this after the car is warmed up. I have replaced the spark plugs, fuel filter, and oil. It has a new air filter, and I also cleaned the Mass air flow sensor. Still doing it! To my knowlege, it has never had a transmission oil change. If I change it, I don't know if it would make a difference. Any ideas?

One thing to note.I have noticed in the last 6 months, a weird shudder in the engine/transmission when it auto shifts down a gear from overdrive.

Also. Engine idle is fine. Purrs like a kitten when sitting. But struggles a bit when under a load(Put into drive or reverse)

Please help!

UPDATE: I removed and cleaned the IAC, and Throttle body. Got it back together. Test drove it. And now it accelerates sluggish. If I gun it from a stop, it'll accelerate fast, but has a difficult time progressing to the next gear. While driving normally, if I give it gas to accelerate, it won't downshift. And accelerate. It's a gradual acceleration. I'm totally lost at this point. I didn't unhook the battery while I did everything (Probably should have) so I unhooked it last night so maybe the computer can reset. I'm going to hook it back up the morning and see how it does. It didn't stall at stops, but wanted to. It still stalled when I put it into drive from park after the car was warmed up though. Any thoughts? I'm now losing hair too!