Personally, I record them through an audio converter and have file folders under a "Music" folder. It auto-alphabetizes them (as long as you're consistent in naming them) and is searchable, and you can listen to the music at any time.

That's a good approach, but it is not as easy as it sounds. To begin with, unless you have really specialist equipment, you have to record 'real time'. So if you have hundreds of LPs at let's say about 45 minutes each, that's (somewhat less) hundreds of hours you need to dedicate to this task. When I tried it, there are programmes that can guess where one track ends and the next starts by detecting silences, but configuring that for unaccompanied solo folk was impossible: tracks were split and joined at random places, and the right gap varied per artist and sometimes per song. So in practice I had no alternative but to sit by the recorder and manually indicate where tracks started and stopped. For, as I say, hundred of hours. But you can't dedicate your life to this, can you? You'd like to do other things as well? Ok, expect to spend several hours a week and so to spend several years to complete the task.