OhKaaay! I’m assuming you’re talking about a ZX6 here, (aka ZZR600) ‘cos that’s what I can talk about – any other bike, then listen at your peril…
The shift shaft can be removed without splitting the cases – open clutch cover and remove the clutch pack, presto there it is, just pull it out (assuming you’ve disconnected the shift lever).
Check if the ‘spline’ is tightly clamped it must be tight. If it is not it will chew up the shifter shaft and spoil your day, so make sure the ‘play’ is not there. The shaft rotates on a tiny needle roller bearing in the casing with a small oil seal. It is possible, but very fiddly to change the seal with the shaft in place, but I’d rather want to know why the seal failed (ie check the bearing).
I’ll put money on it (I’d like to see you come and collect anyway – Heh!) that:
* the play is in the linkages and ball joints. Get onto one knee, giggle the shift lever, and see where the play and movement is…
* the oil coming out there is from the chain…
;-)