Radiator Fans on all the time when battery is connected, but when you start the car, the fans go off

I suspect you may have a bad fan relay or a bad temperature sensor on the coolant.

My 2001 Camry needed a new battery in June 2019. Went through AAA and got a new one. The company’s contracted technician put a new battery in. The following month, I tried to start my car and and everything was dead. No turnover attempt, nothing. AAA came out and jumped my battery. I ran the car for a 40 minute drive and it seemed recharged. The following week car wouldn’t start. AAA said we needed a new alternator. So we got one and installed it. The car started all week right away. The next week I tried to start the car and it was dead again. AAA came out and my brand new battery was dead. Sigh. This time the battery was replaced. Brand new battery in again a month after if getting a brand new battery and new alternator. Then by chance one afternoon, I had driven over to meet a friend. Ran ac and charged my phone on the drive. When I got there and out of car, keys in my pocket, I heard the fans running. I waited a few minutes for them to go off , and when they didn’t, we pulled two fuses out. That seemed to help to turn off the fan. By then it was getting hot during the days and the fans ran every time I used the car. A couple times it was hard to start but was able to finally start it. What’s up? I’ve read and talked two shop guys. I’ve put in new fuses. I’ve wiggled the fuses, which sometimes got the fuses to start the car. It’s really odd. Please help.