Electrical problems with transmission draining car battery and not getting into overdrive

Hello Everyone, new to the forum

Guys I got a problem and I need your expertise. I own a 2001 Toyota Tacoma 2WD Xtracab with would I believe to be a A44D automatic transmission but am not entirely sure. For the last couple of days my car won't start unless I jump start it. So I did the obvious thing and replaced the battery, that didn't solve the problem so I replaced the alternator and guess what, that didn't solve the problem either. When the car is jumpstarted it runs just fine, besides this wild electrical issue the truck is an immaculate condition. Then I happened to noticed that if I shift the transmission out of park and back again it starts like normal and the engine won't turnover in neutral. What the heck. There seems to be something that is draining the battery overnight like a ground fault. Then, a couple of days ago the transmission would not shift into overdrive, I noticed it immediately because of the loud noise in the cabin while I was driving at highway speeds. I closely watched the odometer and never saw or felt the car get into 4th gear (Overdrive) but felt the sensation of engine braking going on as the truck never got into 4th gear. I pressed the overdrive button and it does nothing, it only lights up on the dashboard.

What could be the cause of this issue? The internet points to either a bad neutral safety switch because it won't crank in park or a bad transmission solenoid because it won't go into overdrive. Am I spot on with my assumption or could it perhaps be another issue altogether? Could either of these parts going bad actually cause my battery to drain everynight? By the way, no codes are being thrown on the dash.

If it happens to be either one, could someone point me to a resource or how-to video to replace these parts?