I think one can get the old version from the play store:-
https://play.google.com/store/apps/details?id=com.chess_v2
as chess.com classic.
I have noticed larger data usage I think. Though in my case I do a lot of other stuff which muddies the water somewhat. Your phone probably shows the usage by app somewhere and it would be interesting to compare a "typical" day on each.
On my mobile phone I have 350 Megabytes of Internet per month. I use it to play on chess.com, and rarely for other stuff. With the old app, 350 MB was enough for the whole month. When the new app appeared, I exceeded the limit after 8 days(!), and I can't play on mobile phone anymore...
Is there anything I can do about that? Can I get back to the old version, or anything else?