There can be a few things causing troubles.

The bent derailleur or bent hanger is a consideration, but I'm not so sure you'll get a double shift unless it's really bent a lot.

A worn rear derailleur is viable, and from experience the XT's that were 8 speed versions with the polished aluminum have been more prone to wear and subsequent play / slop. The 9 speed versions which are a satin finish seem to hold up better. You can check these by grabbing the lower cage by the bottom pulley and checking for side play. If (with a straight hanger) while having the system selected in gear 1, check the clearance to the spokes while pushing the cage towards them. I don't know of an actual wear spec, but if it seems to close for my liking I replace the derailleur.

Myself, when I check cables I try not to disconnect the inner wire. My preference first is to select the smallest sprocket on the cassette. Then without using the shifter, move the derailleur by hand while pedaling in the largest cassette sprocket. This normally will allow the housings to be slid free of the frame fittings. This will allow the cable housing to be positioned in some spot mid span and have the end ferrules removed (unless they were swaged) to inspect the housings ends. I find that over time, you'll notice the housing outer sheath will slide back from the cables true end. This normally is the housing failing, flush would be great, a couple of mm's is normally acceptable, after that I believe it's time for replacement. Obviously this does not work for checking the shifter end with taped bars, so the frames end will have to suffice.

With the housing slid back, this is also an easy way to wipe clean portions of the inner wire, and then it can be lubricated before you reposition it onto the frame.

For me, I've typically found shifters are something that works or doesn't. If it will click each gear they normally are ok. Cable housings have given me the most grief, sometimes even when checked as mentioned above, they seem good but still there is a problem, new cables, problem solved. Grit and corrosion can be the other culprit.

If you want to look for a not to common concern, if your frame has the plastic guide under the bottom bracket shells, see if those are intact. Recently on a friends bike the shifting went crazy, similar to this double shift stuff. I installed new cable housings, in the stand it was better, after a quick test ride it failed again. Found the plastic guide had worn through and the cable inner wire was getting caught between it and the frame.

My first pick would be bad cable housings.

FWIW, when I finish a ride, I try to always position the rear onto the smallest sprocket, as this will store the bike with no tension in those cables.

PK