Axle failure on freewheel-equipped rear wheels is a common issue. In fact, that's the main reason why cassette rear wheels (with the freewheeling ratcheting mechanism inside the hub) have replaced freewheel rear wheels for everything other than on low-end bikes.
The axle can certainly be replaced, but the job can be a bit tricky. You'll also need one large (12" or bigger) adjustable wrench - or better yet, a bench vise - to grab the freewheel removal tool. If you haven't adjusted bearings before, it'll probably take you a while to get it done. Make sure that both locknuts are very tight with their cones or the bearings will unadjust themselves quickly.
You say the wheel is fine, well, inspect the bearing races (cups) inside the hubs. If those are shot, the wheel is never gonna run smooth again. Otherwise, you can fix it and make it good again. You might need also replacement cones if the bearings races on the cones aren't smooth anymore.
Otherwise, all you need is a new axle of the proper length, and new ball bearings. Make sure to only use new ball bearings from the same batch - never mix bearings from different batches together.
As for a cassette wheel, there are no cons. Well, other than the fact that it'll cost you, and you'll need a new 7-speed cassette to replace your freewheel, and a 4.5mm spacer since you'll probably be getting a 8-9-10 speed cassette wheel, which only needs a space to work with 7-speed cassettes. Also, you need to check the spacing between your dropouts (O.L.D.) - if your bike uses the new standard, 135mm, then you'll be able to fit any new MTB wheel. If it's 130mm, you'd need to spread apart your dropouts (if your frame is steel) or modify the hub spacing and redish the wheel - making the new wheel option considerably less attractive..