There's nothing worse to getting home, turning on the radio, and being unable to find a clear frequency that isn't full of a bunch of meaningless QSOs where somehow everyone is 599 and no one has anything to say. Are there any designated "contest-free zones", or other ways I might be able to enjoy amateur radio without contests?
Most countries follow the band plans that disallow contesting on WARC frequencies, which include three narrow slices of shortwave frequencies at 30M, 17M, and 12M. If you stick to these frequencies you will rarely, if ever, be bothered by contests.
It's a small slice of spectrum though, so you might simply consider doing what some amateurs do and simply note the days the contests are operating, and plan activities other than amateur radio on those days. Given that most contests are short duration, occur mostly on weekends, and don't occur more than a few times a year, it's not unreasonable to simply avoid affected bands on affected days.
One of my approaches to avoiding contests is operating on the opposite end of the band. During a big contest weekend it can be tough to avoid, when calling CQ, and someone comes back with ID and contest requirement, so the best thing to do is find the extremities of the band. Usually it's the upper end of the band, like for instance on 20 meters, I'll just call CQ in the 14.330 MHz to 14.350 MHz area. It's generally quiet there, and I have found that most call backs are other hams looking to rag chew. Like an earlier answer pointed out, just avoid the traffic by going around it.
Apart from the WARC bands you could possibly try CB radio. I know in the US is usually has a negative stigma attached to it, but alot of my local HAM's are similar to you, as in they hate contests, so they go on CB channel 38 when they can't get through on the other HF bands.
10m doesn't get used very much here, and even if it does, we get horrible QRM, so 11m (CB) is the last resort.
However, if you want to be a 'reckless' ham, you could always transmit voice over the CW and data frequencies. People might hate you for it, but It's not illegal, at least not here (UK).