Hi I'm not sure if you've fixed your car seat but I'm a nanny and broke the car seat exactly the same way, took it out the car and that was it. It took me ages but you need to tip the base and put the back on the wrong way so imagine it as an L when it together take the back of and tilt the base onto where the child's legs hang. Then take the hooks on the back with the head rest facing up and slide it in behind it should just glide in.
I have the exact same swing with the exact same problem, sometimes it will come on and sometimes it does absolutely nothing. It has made me very angry because i took it apart and cleaned the internals up, when i put it back together, it worked fine until my baby wanted to swing the next day, and then it refused to work again. Now the pile of junk is sitting at the edge of my yard after i slung it out the door. My advice, Throw it in the trash where it belongs and go buy a different brand because Graco is TRASH.