If all members of a ring group are busy is there a way for the call to immediately start ringing the next ring group without waiting for the overflow time?

It's possible yes, you need to have the destination in next port and turn on group busy next or all out next, can't recall exactly what it is without checking and I think it might have to be a ucd group but try it as normal first.

I didn't have any luck with that - all out next seems to only work if all members of the group are logged out and with group busy on, i get a busy signal when all members of the group are busy.

Looking at one I have working i have it set as all out next and set as a UCD and sequential.