2004 AVEO bogs down upon normal acceleration from stop

From a dead stop if I accelerate even modestly, the engine usually bogs down, the car will crawl forward very slowly. If I back off the accelerator to a very, very mild acceleration or if I start out by barely pressing down on the accelerator and increase the rpms slowly the car will not hesitate (but it will take a while to get to a speed where I can then press the accelerator for normal operation).

Sometimes if it is severely bogging, it will trigger a flashing engine fault light, which at some point becomes a solid light.

It has a TCM (transmission control module) fault (PO 0700 P) currently; in the past, I also had a misfire fault show up.

check the wires and coil pack.

It sounds like it is time for a tuneup. These are very common symptoms for a misfiring cylinder and can be resolved by changing the sparkplugs, wires, and checking the coil pack as rabbit suggested. It should be a simple, and cheap process that any reasonably mechanically inclined person can manage.