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I have an 88 vert that I just bought, although I haven't taken possession of yet. My brother has had it for a couple of months. It had a bad battery when I bought it, so he put a new one in for me. Ran great for a few weeks (a few short trips to the store,etc.) but now the new battery is dead. Nothing works at all. Sounds like an alternator, but since the car is currently 1100 miles from my driveway, I can't really confirm it.
Do you have a battery charger to charge the battery? Your dash guage should show you what the output of your alternator is. Don't know what you should normally see, but my 96 seems always to show 14.5 when I select the function. My 85 will show something similiar. The voltage rate is managed by the voltage regulator inside the alternator. A new fully charged battery will provide enough electrical power to drive a car without the alternator working but not for long, it all depends on how much drain on the battery like air conditioning on, stereo on, using the power windows, and the cooling fan/fans operating full time.
Not to steer you away from a possible charging problem - but if you have the sport seats there is a constant power supply to the switchs on the side bolsters. These switches get a lot of abuse from getting in and out plus they catch a lot of dirt. Eventually they can become non functional and short out. I had that problem on my 85 coupe and once I located the shorted switch I pulled the fuse until I replaced the switch - actually it is a breaker and it also controls the power door locks too.
.....get a volt meter and measure the volage from the red wire at the back of the alternator to ground...there should be voltage there WITH THE ENGINE OFF.....then, even if you have to jump start the car to get it to run, take another voltage reading at the same place.....
...if you have no voltage at the first test with the car off, you may have a bad wire or blown inline fuse ink.........if you DON"T have at least +13vdc at the second test, the alternator could very well be bad......
Remember - my experience is with an 85 coupe so there could be differences, but there is a breaker (can't remember the amperage) - its a big silver coloured one in the fuse panel. When I pulled it to stop the current to the sport seats it also eliminated my power door locks. Things could be different in your case - but it might be worth checking.