Vox and Fender. Tone Bender, Phase, Flange.
Beatles did about 126 songs. I was in a band 20 years ago that covered over 80 of them. I never tried to get the sounds when with the band. I just used whatever gear I was using.
Paul actually played a lot of the harder stuff in the band. Try playing the lead and rhythm parts in Get Back. If you can do that with the right feel you can probably play most of what they do.
Some of my favorite sounds is the Vox. Tax Man, Paper Back Writer, etc. I'm actually using these types of sounds in my current band.
Early stuff like Ticket to Ride, Help is fairly clean. But not really.
It's actually a huge catalog of great guitar sounds if you cover stuff from 64 to 70. I recommend ball park a few of them and just work with that.