2001 sc2 no reverse

[Rooruby]

I’ve owned my sc2 for about 5 years and had minimal problems with it. I’m the 3rd owner of this car and have all service records. Purchased it as a total loss and repaired it.
Last year I started having problems with it smoking when starting. I worked at a GM garage at the time and was advised it need a head gasket replacement. I didn’t have the money at the time but had another vehicle to drive. So I parked “Jersey” until I could afford the repairs.
Recently I was able to make arrangements to have the head gasket replaced. Luckily my brother is mechanically inclined and willing to do it. So tonight I jumped the car ( battery was very dead after sitting for several months). It didn’t sound too bad and stayed running quite well. I drove it to my dad’s to use his garage for the weekend. (Approximately 7 miles)
When I went to back the car into the driveway, it almost seemed like car was on ice and revved up. I pulled forward to adjust the parking job. When I went to put it in reverse again, shifter moved without hesitation but transmission did not seem to engage. However I had drive. There were no previous transmission issues with this car. Shifted fine on the drive to garage- including up steep hills.
Yes, the battery was completely dead prior to jumping. Yes, service engine soon light on dash. I attributed this to the dead battery.
Any insight would help, especially since I planned on getting the head gasket done this weekend and already purchased everything for it.