I remember having this problem, though my drone reeds have plastic tongues; I think it gets better with time. As for the tuning pins, you might try a tiny dab of wax on the end of the reed; it will lower the pitch a bit and you don't need to be so far down the tuning pin - in fact, this might help the groan problem too.... if the drones are way out on the pins then there's an unusually large cavity in the overall drone bore which may be causing the groan problem. Juolian goodacre tells me that getting the wax to stay put may be tricky with cane; he uses a mix of beeswax and cobblers wax- cheese wax he say is a bit greasy - try testing it with a bit of blue tack any way you'll know pretty soon if it comes loose, and you'll have to fish it out of the bag ...
hope this helps
pete