Swing arm Movement?

[Gnasher]

I've ordered some new bearings, I'll have a look at the pivot shaft when I take it apart when the bearings come.

Don't forget to check the thrust/cover seals, it's bee a while since I taken them off but I'm fairly certain there's a seal built in, if it's buggered they'll need replacing and you don't want to have to do the job again

Will I be ok to take the swing arm off with the bike on the centre stand?

Yes, just make sure you take off any top boxes/panniers etc and bungee/cable tie the front level also be aware with lateral pushing/pulling.    The swingarm should come out fairly easily, if you're forcing anything it's wrong stop and recheck