Which style of AC do I have?

You probably can just look for "non ducted" air conditioners and find yours or another one. They are all the same "footprint" and can be installed in the same 14"x14" ceiling opening as far as I know. There is no reason to stay with the same brand as you have now unless you really like it.

Ducted air conditioners have a thermostat mounted separately from the unit, usually on a wall someplace, and non ducted have the thermostat controls mounted on the ceiling unit as yours seems to have. We have an Atwood "Air Command" ducted unit and have been very pleased with it. Our previous unit, a Dometic "Duo Therm" lasted about 13 years before we replaced it with the present one. The Atwood is much quieter than the Dometic it replaced.
Air conditioners
Barney

Be sure to get a voltmeter that displays all the time, preferably a plugin digital unit. I use this one. Voltage below 108 can damage air conditioners.

CA Traveler wrote:
"Smoke or something" means heat and the source can often be determined visually. Remove all power first.

It didn't smell like melting plastic like I've smelled before when we've had an electrical issue at the power box. We had to have that replaced last year. But this smelled...weird, not like freon though. Like...metallic smelling. That air conditioner runs 24/7 and has been for months. So, maybe we killed it. I figure it's probably going to need to be replaced though & not something that a single part will fix. But I have no idea. I will call tomorrow to try to get someone out here to look at it, but who knows how long it'll take them to get someone out here as I live in Houston and I'm sure they're super busy.

CA Traveler wrote:
Take the indide grill off and you should find a label with brand and model. If not access it from the roof.

Not unusual on hot summer days to have low voltage which is not a A/C friend. Have you been checking the AC voltage?

Ok, I'll have to wait so I can do that as my husbands asleep right now and don't want to disturb him since he has to work tomorrow. The one in my living room seems to be running just fine. It's the bedroom that's the problem. Idk if that information makes a difference or not.

"Smoke or something" means heat and the source can often be determined visually. Remove all power first.

Take the indide grill off and you should find a label with brand and model. If not access it from the roof.

Not unusual on hot summer days to have low voltage which is not a A/C friend. Have you been checking the AC voltage?