I'd guess that there are firms that allow part time agents. You can still work your own hours.
Yes, you do need a brokerage to work for, but that doesn't mean that you will need to be a full time agent or quit your job by any means! And trust me, when you actually start doing deals, you will want somebody that can help you out.
I would talk with as many local real estate companies in your area and just let them know your situation. Most brokerages are always wanting new agents so I don't think you'll have a hard time. Although some may not want part time brokers, but that is fine!
I couldn't be happier with my brokerage I work for. The owner was extremely accommodating and helpful when I was getting started.
My only advice would be to be to think about your commission split with the brokerage. A lot of your big names with the big offices will take a LOT out of your commission, and if you are only having your license to help with your investing, I would look for a place that gave you a higher split so that when you do complete a deal, you get most of the money.
Good luck on your adventure!
In North Carolina, yes. I don't know about Washington. Although, I'd suspect as much.