Plumbing – shut off the cold water supply to an oil furnace while away on vacation

If you have no separate shut off valve you should wait till the local water authority can come and temporarily shut off the water at the meter (which is presumably near the street).

I see no problem with shutting off the water supply to the water heater as a short term measure to curb the water waste from the leaking faucet in the shower. It would be a good idea to turn off the water heater during the time that the water supply is turned off.

After you turn the water supply back on the only issue you will have with the water heater is that some air could have fed back into the tank. All you have to do is to open the hot water valves in several of your sinks and let them run for a short while until the water runs clear without any bubbles or spitting coming out of the faucets.

I think what is leaking is the drain. Does it look like the object sticking out to the side in the below picture? If so, this is used to drain water from the house side of the supply when the valve is off.

It sounds like you over tightened it in the first place which started the leaking. If you've broken only the cap, you might be able to find a replacement. If the stem and threads are damaged, long-term you will need to replace the entire valve. As a short-term fix, you just need to block it some how. Marine epoxy is handy for emergency plumbing repairs.