i encourage you to keep at it! these can be a pain to fix but it's very satisfying once you do.
if all the other measurement ranges (ohms, volts, etc, and you've verified 10M dcv input impedance) are working than I'd probably start with the piezo driver:
if you look at the schematics you'll see U4 pin 15 (signal p23) goes to the base of Q9 to drive the beeper... is the p23 signal turning on/off when you change the continuity button? if so, q11 could be bad. if not, check the other piezo driver bits (see section 2-18 in the service manual)