I would out the straw on top, gives you an absorbent layer and saves straw. I put a bucket of shavings down where mine wets, and it really keeps the bed dry. If I don't, it's a mingling mess in the morning.
When I swapped from wood pellets to straw I just left the old bed underneath and cleaned it out as it got soaked.
I'm loving straw having not used it for probably 20 years - I'd never have gone back to it if it wasn't free this year but now I think I'll stick with it!