@Chris G-F
Welcome to the forum and good luck with your move!
If you get a place in one of the main cities (Sofia, Plovdiv, Varna, Burgas) then you could remain car-free if you wish. We live in Plovdiv which is a great city, and relatively small. We live in an upscale district next to Youth Hill and the Regatta/Maritsa, but it's still only a couple of klicks to the centre. It's basically a 5-Minute-Neighbourhood in a 15-Minute-City, so it's very easy. We still have the car, but usage is very minimal as I'm on a micro-mobility kick with my escooter. I use it to get around everywhere as Plovdiv has a great network of walking/cycling paths, so I'm 2 minutes from Kaufland and 10 minutes to downtown, with zero parking aggravation.
You should be fine on your US driving license for a while, so no panic yet. I'd renew it (if it's easy), and you should consider swapping it for a Bulgarian license if you plan to live here indefinitely. You can swap at your local KAT, but go soon before your license expires.
You can't have a Bulgarian driving license or a Bulgarian registered vehicle until you're a legal resident of Bulgaria. (Local bank accounts will be easier to open then too.) You can do this by Family Reunification as the spouse of a Bulgarian citizen, and I suggest it should be one of your first priorities. You need to check the rules for this as you might need a D visa from a Bulgarian embassy before you arrive, or you may be able to apply directly for residence in Bulgaria.
I very much doubt you can get a car loan/lease as a non-resident, and maybe not as a new resident. But no harm in asking, if you visit a dealer. I'd guess it would be a lot easier if your wife tried to get a car loan/lease in her name, and she can immediately register it too. I doubt car leases are as big here as in USA. They are (financially-speaking) a horrible deal, you'd be better off buying a used vehicle for cash. But I'm an old-school frugal shopper, Ebay bargains make me happy. :-) In the short-term you might consider a long rental (from somewhere like Top Rent A Car in Sofia) while you hunt around for a property and a vehicle.
The other option might be to try for a car lease in Austria as you're more established there, and drive it in Austria and Bulgaria.
In terms of how it works, it's very similar to most places. You schlepp around a bunch of dealers looking for your ideal vehicle, then you haggle over the price/extras and whether they will/will not extend any financing. The transfer of ownership requires a notary in Bulgaria (so buyer and seller formally sign the documents). You have to go to KAT (the traffic police) to get the vehicle registered (if it's new) or transferred (if it's used/already registered). As mentioned, you can't register it until you're a legal resident, but your wife can as she's Bulgarian. (If you insist on getting one for yourself sooner, you can form a Bulgarian company and register it as a company vehicle.) You'll need a helper/translator (or your wife) for all this stuff.