It has been a long time since I've had an MGB (something in the works right now....) but there were two (2) driveline "fixes" that I would do almost immediately.
1) was to replace the four (4, 2-flat, 2-conical) thrust washers and the roll pin in the differential.
2) replace the laygear shaft, needle-rollers, thrust washers & the four (4) synchronizer rings in the xmsn.
Keep in mind that it's been almost twenty (<20) years since I left my last three (3) MGBs (1-Tourer, and 2-GTs) in my ex-garage...) but the trans tune up was around $100.00 and the parts for the diff were a fraction of that.
When the laygear shaft wears, the clearance increases between the constant-mesh gears, leading to much more expensive repairs. If the unit had an overdrive fitted, I'd fit new thrust washers and operating piston O-rings.
The differentials are pretty strong, but with too much slack on the spider gears__and too much horseplay__it is possible to shatter one of the floating spider gears; BTDT. With no other damage, robbing the four (4) spiders from a loose axle and the car (in that case, a Blaze/Navy '73 BGT) was again moving under its own power. It's an easy job, so my recommendation is to get on with it