You don't really have to change the way you do things. The bow's you're shooting have "recurve " grips, even though their limbs are more hybrid r/d. Should feel pretty much the same as your recurve, though perhaps a bit larger, smaller, etc.
If you pick up a longbow with a straight grip, you can still do pretty much the same thing, but you'll need to lower your wrist, i.e., be sure to get the balm of your bow hand down on the grip. I let the longbow find its proper place in the web of my hand and palm,and then draw the bow with a grip that cradles, but doesn't squeeze the riser. I do wrap all four fingers (loosely) around the riser though.