As I have said many times, "I have been at this a long time." There are probably as many ways to do this as there are folks trying.
That said: I run the tap water so it is moderately warm. I hold the leather under the faucet until there are no bubbles emitting from leather, turn it over to opposite side and do the same. Use an old terry cloth towel or old clean T-shirt laying it flat on table, lay the dampened leather on top of the cloth and roll the towel and leather together. Leave overnight. When you unroll it next day it will be damp and a little darker in color than when dry plus very pliable. Lay the leather out on your stone you use to engrave and stamp. Allow the leather to dry enough you can see it begin to change back to original color. It is ready to work. Naturally it will dry out as you work your stamping. Use a large cellulose sponge that is wet, not dripping, to go over the leather piece. This will replace some or most of the moisture you started with. Continue your carving or stamping.
When I am finished with the engrave I make sure the piece is laying flat and place the towel over it overnight. You can do what ever finish you wish at this time plus the leather will take an impression with a personalized stamp with ample pressure.
Hope this helps. Remember one very important thing. Take your time with casing the leather. Do it correctly and you will never regret your results.
Photo is of a piece I carved over 40 years ago after using above tempering method.
Ferg