Here is my 2 cents worth...
I assume that you mean lateral movement (up and down/side to side) and not radially (around - normal rotation)
Some lateral movement is unavoidable, but if it's over... say 1/4", it's too much.
The bushings in the tailshaft housing and the rear of the transmission case are probably shot. If you dog it and have a V-8, they are probably gone. If these are worn excessively, it can cause a variety of internal ills that slowly creep up on you.
You can try replacing just the tailshaft bushing, remove it and take it to a machine shop to be pressed in, unless you have a press. The bushing is about $2.00. Get a new shaft seal and housing gasket as well. The other bushing requires total dissasembly of the tranmission to replace.
Nothing explodes when you pull the housing. Just be careful and note where things come from. Depending on the year model of the trans, there may be a sleeve and pin over where the governor USED to be. Don't remove it or loose the pin.
Don't break the speed sensor, and be careful to remove all old gasket material before reassembly to avoid a cocked housing - and further wear. Lube all part contact points with ATF before reassembly. Because it is aluminum, retorque the bolts evenly and in steps, just as a precaution. Put a little grease around the inside of the seal lip before inserting the driveshaft - not so much that it ends up inside the tranny.