This is the way.

Of course it can be done, check your web server logs.

If you are using GET requests to send search queries to searxng, what you searched for will show up in the logs as

2024-10-31 123.321.0.100 /?query=kinky+furry+pictures

If you use POST requests the server admin can also easily enable logging those.

People hosting searxng can absolutely see what you searched for, along with your IP address, user agent string etc.