My guess is yes, animal control probably can legally take away the chickens you are illegally harboring.
You can try talking with them (or better put your request in writing) and requesting an extension of time. If you have proof that you are moving they may give you an "extension" on your time frame.
Or you can let them take and hold the chickens, pay the fine and get them back once you've moved. Not what would be idea... sometimes we get ourselves into circumstances and have to deal though.
For future reference... there are always consequences if you get caught doing what you know you shouldn't.
Maybe you have a friend or family who could temporarily house them while you are moving. Or you might even rehome them and get some new chooks once you are done.
Don't know if it will change anything but you might be able to move them to an indoor kennel temporarily.
I'm against people just breaking the rules because they think thjey can get away with it.
But I realize we all have reasons why we do what we do... we are here to support each other at BYC, and help find solutions and maybe make better choices if we face them again in the future.
Hopefully some of my suggestions will be helpful. Moving is hard. Good luck and best wishes!