I'm going out of town for a month. Should I shut off water and draing water heater? I live in a cold climate and tank is in garage next to the furnace.

I have a Toto toilet that I have been using for 6 years.A few days ago, I noticed that the level in the bowl after a refill was very low. I checked the fill tube in the tank and it was attached and working properly. If I add water to the bowl, the level will stay until it's flushed again so,I've ruled out a leak. It seems that whatever water is fed into the overfill tube while the tank is filling is draining out of the bowl as it tries to fill.
I have flushed and refilled with the bowl fill tube connected and not connected and there is a difference of the amount of water in the bowl. I think I have checked everything I could think of. Any ideas?
Thank you so very much.
Ken

How cold is cold? Is the garage heated? I would definitely shut off the water main but draining may not be necessary if this area never freezes