I remember my M37 as having a synchronized gearbox. They were supposed to from 1951 onward.
I just used my M37 as shipped and actually didn't mind the gerabox at all so no first-hand knowledge of swaps. Here are some things I found though...
The M37B1 transmission is liked better than the original M37 gearbox. The G741 Forum might be the place to check for details and easy swaps. I gather there are various gotchas about brake pedal locations and the like so your options might be pretty limited if you want something close to drop-in.
I think what you want is here, identification and listings for both the early and late NP gearboxes as used in the M37 and similar:
drive.google.com
If you want to keep it as Mopar as possible, I'd check into various New Process gearboxes, I think both the NP420 and NP435 would likely fit with some work and are pretty likely to just bolt-up on the bellhousing side at least. I know they came in a variety of splines and input shafts though. Not fully synchro'd but second and up is a good start. They were replaced with the fully-synchronized NP4530 and NP445.
Might be an answer for you over here using Ford gearboxes:
hello all, I have a 1933 Dodge DP 6 with a 1953 Dodge 230ci engine, the stock transmission in the vehicle is a 3 speed with no-syncro's and not very good shifting sounds. it either must be rebuilt or will be replaced with a new updated transmission. I am not familiar with what to do in this scena...
p15-d24.com
I am looking into swapping out my 47 dodge trk 3spd for a 5spd T5 trans with the stovebolt adapter. I wanted to know which trans to use and if there is any modifications needed like redrilling the bell housing. I was looking into an early model S10 5spd with speedo cable. Thanks, Greg
p15-d24.com
If you want to run a Chevy gearbox and don't mind spending some $
Wilcap engine to transmission adapters for Ford Flathead Early Hemi GM Chevy
wilcap.com
Good luck and let us know how it goes.