1996 LSS crank-no-start problem

Need Help! 1996 Olds LSS cranks but doesn't start. 89,000 miles. Seems to happen when outside temp is in 20's. Driven daily, all traffic conditions. When it cranks, it sounds normal. It dies immediately. No dash lights, no clicking, nothing. Clock resets to 1:00, radio stations stay set. Have to wait a few minutes before trying again. If I try to early, no sound, no lights, not anything. Runs great once it starts. Left turn signal also acting up, doesn't blink consistently. Tried a fuel cleaner, no difference. Owned it 3 years with no issues at all. Any advice for my husband to try?

Here is what my 96 Achieva is doing,


It was very cold out 15 degrees. I remote started the car and it did not start. I tried multiple times till the battery was near death. I tried leaving the key in the ignition for ten minutes to reset the passlock, didn't work. I tried to disconnect the battery, didn't work. I waited several hours and it started. I ran the car on the parkway and it died while going around a bend. All dash lights came on, interior lights came on then everything blacked out. The car then started back up while hesitating. The security light stayed on while driving. When the car was shutdown it started up but the RPM's immediately jumped to 3000 then slowly dropped...

How is your car doing?