it would help to know jurisdiction/market area because that really affects the options, especially because that's a rapidly changing landscape right now. some places, prepaid SIMs + cash work well enough, other places it's better to go with giftcards or prepaid debit cards etc. yet other places there's definitely no good options to do it. it takes a lot of effort, and you have to remember that you need to consider that any app or service you log into on that phone may end up identifying you against the SIM card, so you need to be really careful in your usage! i hope you find an option that works! :)
It might be worth looking again at pay-as-you-go. Here in the UK at least, there are some 'bundle' plans that are not actually that much more expensive than a contract. Topping up with a voucher once a month is a bit inconvenient but not too bad. You
I don't know to what extent the phone companies log and share your usage (if you're in an English-speaking country it's likely quite extensive) but I would imagine they could still identify you probabilistically by your patterns of use, including which towers you connect to.
Using a VPN/proxy/TOR would at least limit the number of IP addresses they see you connecting to but that's off topic a bit.
i've been using prepaid sims for over a decade and a half now and it works okay for me since i only use a voip number that forwards to my phones.
every major carrier i've used in the united states have required some sort of identifying information like a state issued id or credit card; maybe the smaller/re-sale carriers will let you use cash.