Regarding the DG:

Sounds like a mechanical interference issue, which could account for both
the "snagging" on the 270 hdg and the bug being "frozen". The DG may only
need repair rather than full overhaul. Take it to a competent instrument
shop for diagnosis.

If it does need overhaul, whether to do it or not, in my opinion, depends
upon the quality of your unit. Some of the newer "cheapies" are not worth
overhauling while older, more robust designs are. Again, a good shop should
be able to advise you.

Regarding the DME issue: If your DME doesn't read to tenths of miles but
instead rounds to the nearest mile, the discrepancy may not exist. If the
DME read exactly 3.0 miles from the VOR and ATC told you 4, that could be
just a somewhat sloppy guesstimate by the controller. My understanding is
that they can bring up range rings on their new screens, but if a controller
is busy he/she might just "eyeball" it. In any event, I wouldn't worry too
much about it. If the DME indicates distance it is most likely functioning
properly. Measurement errors are conceivable but not common. Of course, as
you suggest it will be simple to do a check against your GPS

--
-Elliott Drucker