Is it a problem that my toilet flange is 3/8" below the surrounding tile?
A double wax ring will probably work but IMO the better option is a toilet flange extender.  It is just another flange ring that sits on top of the original flange to get it up to the correct height.  Then you install the wax ring and toilet as usual.

I don't have any experience with this particular kit but something like this Toilet Flange Extender Kit.