Yes, it does mean we’re short on time. Literally, being in a crunch means being under some pressure, timewise. Alternatives would be just „we‘re short on time“ or „for us, time is of the essence“. , this latter being rather formal sounding, more written speech than spoken. We’re in a crush is very common and it’s colloquial, so yes. Very common. Used all the time, usually in the spoken word.

It's usually used in business referring to a project that needs to be completed.