From the way your post is worded, it sounds like you may have two different problems going on.
A) Last night I heard a strange noise coming from the wall between my room and my bathroom. It sounds like there is a stream of water constantly running.
The first place I'd check is the toilet. It could be that your fill valve isnt shutting off completely and may need to be replaced.
B) This morning I turned off the water at the main and turned on the sink to drain the water from the pipes. I wanted to see if it was water I was hearing. The sound stopped, but when I turned off the sink after a few minutes, the sound started again. I turned on the sink again and more water came out (for some reason, it never stops)
You say you turned off the water, turned the sink faucet on to drain the lines, turned the sink faucet back off, turned it back on and there was water in the lines again. Right? I'd check my main water shutoff valve. If it is an old gate valve, it may have failed in the fact that you cannot get the water completely shut off.