Indicator light issue with 1996 Chevy Cavalier

I have a 1996 Chevy Cavalier with 78100 miles that developed an unusual issue after I replaced the battery cables. First let me say that I was extremely careful not to short anything and follow the proper procedure (negative and then positive, etc). Unfortunately, something still went wrong because there is a gremlin in the electrical system! When the key is in the lock position, nothing on the instrument panel is light and other normal things occur, such as the “key chime” with the door open, the dome light coming on, etc. With the key in the OFF position, the indicator panel lights up while going through the tests. The Air bag, Parking Brake, Seat Belt, ABS, Theft System, Battery, Oil, Coolant, Check Oil and Check Gages come on. Only the Coolant, Check Oil and Check Gages light go off after a few seconds. The rest remain lit. Then if the key is moved to the RUN position, the “chime” sounds, the ABS light goes off, and other lights go off, if that part is disengaged or engaged (i.E. Seat belt, parking brake, etc). So, that leaves the Theft System, Battery and Check Engine light stay on. Now, here’s where things get weird. If I start the car, which doesn’t start or idle rough, or misfire the indicator lights begin to flash on and off. For example, if I turn on the lights, the daytime running light indicator will begin to flash and then the ABS and Theft System light might come on. The good news is that the indicator lights appear to be the only thing malfunctioning and they don't affect the function on the car. The bad thing is that nothing seems consistent where a certain lights comes on or goes off at a certain time.

I checked the voltage of the battery at rest and it is 12.57 V. With the key in the RUN position, the volts drop to 12.38 V. Then when the car is started, the volts jump to 14.27 V. That’s normal to me.

Anyway, I’m not sure where to test or troubleshoot next and I don’t want to bring it into the dealer just so he can charge me $75 dollars labor to replace a relay!

If anyone has any suggestions, information or advice, please reply. Also, if any additional information is needed, please let me know!

Thank you in advance!