It sounds like the consensus with the group is to use a search-specific add-on, and they're probably right (Bob would be the guy to know!).
If you did decide to go the getResources route, you'd basically be nesting the getUrlParam inside your getResources call. So, it would look something like this (I highly doubt this would work as-is, but should show you the idea):
[[!getResources? &tvFilters=`searchTermTV==%[[!getUrlParam? &name=`searchterm`]]%` ]]
In that example, "searchTermTV" is the name of the TV that contains the data in your resources, and "searchterm" is the query string parameter you pass through your form.
You're right, though, that this could create ugly URLs. The alternative would be to submit the form data using $_POST and grab the submitted query that way, but that would probably require a custom wrapper snippet to grab and sanitize the values, and pass them on to the getResources call.
But again, just a proof of concept. An actual search add-on is probably better.