Before I say anything, I don't care if you use a meter or not. Also, it's NOT a whole new ballgame. Unless you're shooting Velvia on a very contrasty scene or something. Portra 160 is ridiculously forgiving on highlights, you've got some range to work with.
But, I use a meter with a polarizer, even when I don't use a meter. By that, I shoot through the polarizer to get a feel for it, how much it really cuts and where. Then I just use that as a correction factor if I'm using the thing. I do that with my color filters for B&W too.
When I started back into film two years back I was into contrasty scenes. Landscapes, sunsets, reflections on water, using GNDs -- stupid hard to just "nail" every time and I had no through the lens metering. I pointed my meter everywhere, even if I wasn't shooting, just to get a sense of the range of the scene. After a while I could just take an incident reading as the light was dropping off and know all the rest.
You might do a similar thing. Use the meter where it's useful, then you'll be good and can leave it at home. Or just keep notes as you shoot and correct on the next roll if you're way off. I'm betting, with your experience, very little practice and you can nail overcast 5.6 way faster than I did.