98 Saturn SL Won't Start- New Battery

Just got my 2nd battery of the week and 98 Saturn SL (69K) still won't crank on cold start.
With 1st battery it was like the theft detection system turned all electricity off. But for the past 10 days, putting into Neutral, playing with ignition, or going outside and locking and unlocking with Transponder I could eventually get electricity and engine would crank and start. But it was like all electricity was lost. Radio pre-sets and time were lost. After car was started it would run fine and re-start for the rest of the day but sometimes had problems the next day, sometimes it was fine though. Not too cold in San Francisco, just wet but the Saturn is inside the garage. Took car into my favorite shop, on their advice since battery was still under warranty;only 7 months old had new battery replaced.
With 2nd battery, which tested out fine and charging system checked fine after installation, on second day car will not crank again. Electricity seems very weak. Playing between N and P, I can sometimes get radio to show light but electricity seems too weak to use transponder to arm and disarm doors. Interior dome light seems very weak.
Any guesses to what the problem is? Is there an anti-theft electrical kill switch which can be turned back on easily? Is the ignition going? Would you aim towards electric shop or dealer? Not the best start to a new year. Single mom and need the car working. Advice appreciated. Thanks! A7

basically i would give it to saturn and pay them 41 dollars for a diagnostic test. see what they say and get back to "us" on it