New 10-speed chain and cassette is probably wise with a 10 shifter, but there's the slight chance the new chain won't love the cranks.
BITD when 9-speed D-A first came out there was a cable trick to make it work with an 8-speed wheel: you route the cable at a right angle on the closer-to-loop side of the derailleur fixing bolt instead of behind it (Google for Sheldon Brown pic which shows this).
Maybe it works to make a 10 shifter work on 9, or maybe you buy the other stuff after. Couldn't hurt to try!
Edit: Here's the link. Scroll down to "alternate cable routing".
https://www.sheldonbrown.com/drivetrain-mixing.shtml