Yes, the '05-'06 Tundies all got the A750...Try to stay away from the '00-'02 Tundras with the A340 auto, unless the vehicle condition is excellent, with low miles, good price, and you dont plan on towing. The '00-'02 A340s are known for having weaker planetaries, IIRC, due to a parts supplier design. Toyota switched suppliers for the parts for the '03-'04 years, and it seems to have fixed the failures. If you DO get an '00-'02 and the tranny does go, just replace it with one known to come from an '03 or '04...They are total plug-n-play, bolt in swap.
Also, inspect the cooling system (especially inside the coolant reservoir) for an thick, orange-pinkish sherbet looking goo floating in the coolant or stuck to the inside walls of the reservoir. Early 1st Gen Tundies have been known to have issues with the transmission cooling line fittings corroding in the bottom tank of the radiator. If they go, it ends up mixing coolant with ATF...Not good for the tranny, especially.
If you are in a cold weather state, inspect the Tundra frame VERY well...Especially the rear crossmember that the spare hangs from...There was a big TSB/Service Campaign for frame replacement or frame coating/crossmember replacement for rust issues.
I came from a '99 reg cab 3RZ 5spd 4wd Taco to a '00 Tundra access cab V8 auto, and the difference in economy was a large shock. Handling, not too much different, though the Taco was a bit quicker in manuevering...Not surprising since IIRC, the Tundra has a good 5'-6' longer length than my old Taco. Also, the tundra feels more planted, and doesnt get loose in the rear like my Taco did. I was always spinning/chirping tires without even trying, or getting on it, from stop lights especially on wet roads. I HATE automatics, and the Tundra is only the 2nd I've ever owned. (I've owned around 20 vehicles, and all but 4 were Toyotas)...i REALLY miss having at least a 5spd manual in the Tundra.
Oh, and if you're not already, sign up on Toyota.com as an owner. Then you can enter the VIN of ANY Toyota you're considering purchasing, and see service records for anything ever done at any Toyota dealership....A nice little addition to supplement a Carfax, or to verify any service/maintenance, recalls, TSBs the seller says they have had done by a dealer.