A lot depends upon where you live and the lifestyle of the people around you, but here's some ideas that I use......
.....Raise your own. I don't get excess livestock that I can't either home use,
sell, or trade, but other farms do. So example shows you could raise your own or cull your excess. Since you find the slaughter and butchering objectionable (I do my own because it is far more humane) you could arrange with a neighbor to do that for you for either cash payment or share part of the meat.
.....Slaughter waste from others. From time to time I get bags of slaughter waste from neighbors and friends. They don't want it and my animals can eat it.
......Hunters' waste. I trade for hunters' waste. They drop off a
bucket of offal and trimmings in exchange for veggies from my garden or eggs.
......Road kill. This s a great source of guilt free pet food and a means to practice butchering. I take in free killed roadkill from people who bring it to me. Yes, I'll pick it up myself when someone texts me with a sighting, but frequently it shows up at my gate. I trade roadkill for veggies, fruits, eggs, good meats. Roadkill around here is often feral pigs, goats, occasional sheep. Plenty of mongoose, pheasant, feral chicken, and sometimes a turkey. Those last four are usually pretty flat.
......Farm losses. Livestock sometimes dies unexpectedly and needs to be dealt with. As long as the cause of death wasn't from illness that would affect humans or my livestock, I'm willing to pick them up. Old age, physical injury, lighting strike, hit by car, illness, killed by dogs, impossible birthing problem, etc.
.....supermarket and restaurant waste. The pick up contracts are pretty well sown up in my area but I do have one market that I can get about 10 lbs of discarded meat from weekly. Plus the
local churches frequently host free meals and festival dinners. By providing them lidded cans and timely pick up, I can get all sorts of waste including meat trimmings.
......trapped and otherwise culled vermin. I live where we have significant rodent problems and an excess of certain types of birds. Care needs to be taken that neighbors aren't putting out poison for vermin. I trap dozens of rats,
mice, and mongoose every week. They go into the chicken feed cook pot. I sometimes have requests from neighbors to come reduce the numbers of pheasants, mynah birds, and doves. People are happy to have me come trap the feral roosters and excess feral turkeys. You have to take good precautions not to accidentally injure pet cats and dogs though. While I often find cats in my live traps, they are released unharmed. But it's super important to be a responsible trapper.
Hope this gives you some ideas.