99 Sunfire - Transmission problem? Electrical problem?

Been having this problem with my Sunfire for about 4 years now. I've been to several mechanics, including the dealer that sold me the car over a decade ago, and no one seems to know what's going on.

This problem only happens in the summer. After driving for a little while, every time the car shifts gears, it jerks really noticeably, almost like the transmission is shifting too soon. If I keep driving it, eventually it will hesitate when I step on the gas. If I stop, and then try to accelerate, nothing happens. Sometimes my battery, oil and engine light will come on when that happens, and I lose power steering. The only way I've been managing this problem for so long is pulling off the road when this happens, and shutting the car off. It starts up again fine, but when I shift into drive it will bang into gear and carry on like nothing happened. At least, until it happens again a while later.

Does anyone know what's causing this? I thought it was the speed sensor, but after cleaning it, it still continues. Since it was only happening in the summer, I then thought it might be linked to my a/c, but it still happens even when I have everything shut off. I took it to a mechanic, and was told it was the transmission. So I had that looked at ... still no change.