Things seem a little confusing
If I understand, I say yes move the fish (if they are showing no signs of disease) to the 20g - though
how many fish are we talking about, and is the 20g cycled now?
Cut back on feeding if not done so allready.
Have you got test kits - what are the readings?
Once all the fish are moved you could also add the filter (easy if an internal) from one of the 10g to the 20g if the 20's filter won't be sufficient by itself. Then concentrate on getting one of the 10's cycled to rehouse some of the fish.....Hope this makes sense
unless someone else wants to help with this situation.
hth and keep us updated