We self-host nominatim as part of the iplocate.io pipeline. It works great, but the requirements are pretty heavy for something to host casually.
An in-between for OP could be something like opencagedata.com, which is still a third-party API but an order of magnitude less expensive than Google. (not affiliated but have previously explored the service)
It can be self-hosted, with constant replication. There's also Photon which is a cut-down version of it: https://photon.komoot.io