Hi, welcome to BYC!
A dilemma many of us face. My hubs also does not participate in processing. My daughter and I did/do it.

We only do one or two here or there as needed to maintain good flock dynamics. Took us a long time to get there... we openly talked about it and knew that was where we were heading.
Yes I keep them all together until we process, regardless of breed... and we even eat our extra Silkie roosters. We did it our first time after watching a LOT of videos. It went horrible, but we got through it and learned A lot. We still don't enjoy doing it, but having the skill to confidently help out a bird I care about if they get in trouble has been invaluable. I choose the cone method as being the least dangerous to myself.
You can advertise on craigslist in the farm and garden section for someone to process. Or you can call all the local butcher shops and inquire. You might post a note on your local feed store posting board or facebook if you do that. I did find someone to process for me. It cost me $5/ bird and is well worth it. I have bartered for 30% of the meat since it cost me $10 to raise and $5 for processing that $15 total per bird, so I figure a 30% value. Most people who are willing to process, already have their own birds and not as willing to barter for meat. But each situation will be unique. My processing guy prefers broilers and I raise heritage breeds. They are MUCH different birds when prepared for the table. It takes weeding through replies to find someone who is actually good to work with and also it takes time. He did not reply when I advertised, but much later when he was looking for himself a rooster for his hens and wanted to barter. I had already found another person who was willing to do it for me for $4/bird... but they were 90 miles from my house so it would have been an all day trip that included transporting live and dead birds. You might even contact someone who is selling their broilers and ask them if they might process for you. I just apologize in my original message to them if this counts as soliciting.
Generally speaking, age will depend on breed some. Cornish cross can be butchered at 6-12 weeks. The heritage breeds not usually before 16 weeks to make it worth the meat. Many people wait until around 20-24 weeks, especially for birds like Orpington or Faverolles. So far we just process on our schedule when it's convenient for us regardless of age, but not usually before 16 weeks. And our first round boys were easily 7 months old before worked up the courage to proceed. Since my daughter and I only do a bird or so at time when needed... we skin instead of plucking.
Yes, it was large game processor (deer, elk) that hooked me up with my first good lead for chicken processing, mentioned by the other poster.
I was afraid I would be vegetarian before it was all said and done. But instead I'm just well aware of where my food comes from AND the quality difference. Our consumption has become more conscientious.
Good luck finding someone AND learning how to handle it yourself if need be.
It's a worthy adventure.