As ever with fleece, I seem to fly contrary to everyone else...
I don't like combing squeaky clean fleece, it's too fly away. So I comb either unwashed (just cold soaked) or
lightly washed fleece (that still has some grease in it.)
You do need to be in warm, though, yes, so that the grease isn't solid.
To clean off grease from carders or combs, put some meths on some waste fleece and and card or comb that. The tines will come up squeaky shiny clean
Which combs did you get?