My rear cassette keeps shaking back and forth when coasting, especially when I have the clutch off, any ideas what could be causing it or how to resolve the problem?

what cassette, what hub?

It may be your hub.. A lot of cassettes have a slight bit of wobble..

Knackered freehub bearings would be my diagnosis. Particularly if it is a cartridge bearing hub like Hope. The good news is bearings aren’t difficult to source and swapping them is not particularly difficult. Don’t worry too much about drifts and the like, just use sockets of the right size and a bit of common sense and mechanical sympathy and you’ll be reet.

How hard is it to change the axle?

My bike did this last week. Wondered why I’d suddenly started dropping the chain, looking down I could see that the top run of chain was going slack when freewheeling meaning that the freehub was sticking. Bearings we thought.
Took the wheel off when I got home. Freehub and cassette fell off. Snapped axle.
10 minute fix. Job jobbed

Where can I get a new axle, is it a quick fix?

As you havent answered the question of what hub you have, I would suggest going to a local bike shop 🙂

Stans hub, sorry I missed that before