Sounds like either the positive or negative output transistors are blown.
The output transistors are probably "open" rather than "shorted". If they were shorted you'ld have DC on the output all the time, and the speaker protection would never come on. With them open, it comes on at zero volume. As soon as you bring the volume up, the waveform is lop-sided, so you get DC, and the output relay kicks out.
Whether they're shorted or open doesn't really affect the cure, you'ld need to replace them. Its just a question of determining which. A meter with a diode tester, or a scope will help.