Just as you say mate...
Dry cut them on wet saw or grinder...Wipe over with a dampened sponge and leave to dry....then seal as you lay( if you can work clean)....I use lithofin stainstop for travertine ..dries quickly....Rapidset adhesives and grout once set....can be done in one day no probs......

.... just make sure it is clean before sealing....
p.s. 3 mm joints will look good....