The OP said it is a Carrier unit. You can copy the model number and serial number from the air handling unit in the garage. Then, you can go to the Carrier website and it will show you the exact status of the warranty. Most units in The Villages come with a 5-year labor and 10-year parts warranty. But, if anything is still under warranty, the home warranty company will not perform work on it.

Warranty.

Broward has a good reputation in the Northern Villages. They may be having parts issues after 2 hurricanes. I expect they'll make good. Just get everything form them in writing.

Run to small claims court. Just filing the claim will probably get them to cover it immediately.
Can you see/reach where the leak is? It would be a shame it that got worse. Nudge nudge wink wink

My OPINION. It is a matter of TRUST. You can call another company for a second opinion. You will find they will bill you for a service call-roughly $100. For you to make that decision. They added Freon-the coolent gas. You should have gotten a written report. Measure xxxxxx added xxxxxxx.
Far as a leak. Like everything today, it is government regulated. IF, it is leaking beyond limits they cannot add more coolant. I had read average life is/was 14 years in the villages. Reminder you use it all year cooling and heating. Gossip, perhaps? They don't make em like they used to. A tech told me it is down from 14 to 11. A replacement will cost like 8,000 plus.

Seniors vs Crime
 

Seniors vs Crime in The Villages
Seniors vs Crime may be able to help. One call from them on your behalf can work wonders and they are volunteers - no cost to you. They are a division of the Attorney General's office so that's what they say when they call. Worked for me in several instances. Good luck!

The Federal law is only for units with 50 or more pounds of coolant. OP’s unit is an eight pounder.

Carrier, I've read,is no longer made in the US, it is made in Mexico. Trade wars? There was an article in Barrons,avail on line, that explains it well in plainspeak and they are not selling you something.

Broward Gold would have replaced it

Thanks, 12 years old. House built in 2011.

I tried to buy the Gold, but they wouldn't sell it to me.

Thanks!

I just checked the warranty & it was for 5 years. Expired in 2016. Thanks for the info.

Insurance is for things you cannot afford to lose. Home, car and life.

Everything else warranty is huge moneymaker for the folks doing the warrantying.

Yes, and the home warranty companies are not in it to loose money. Why would they?

There is a very high probability if it’s a small leak that it’s on the indoor coil. Adding refrigerant to bring it back up to the proper level will make it work but might only be for a short time. Running a system that’s low on refrigerant can do damage to the compressor. if you have a leak, you really should try to get it resolved. Replacing that component, (coil)would be an option. Depending on the age of the system, it might be wiser to replace everything. Typically home warranty companies would not replace that coil. in my experience they will only replace components that are rusted. I would get a good HVAC company to check it out and they should be able to look for a leak and once they find it can give you options. Good luck