I live in the PNW (Pacific NW), so hiking wet is something it's best to get used to.
I think having the right set of expectations and mental model is key. When it's raining and/or cold-humid out, "dry" is a relative term, and not something to stress overmuch about.
Ultimately, it's about temperature control and not dampness. If significant rain/wet is likely, then I do have some at least minimal "sleeping clothes" (I sleep in my hiking clothes otherwise). For me that might be silk longjohns and a capilene 1 t-shirt, plus of course some sort of dedicated sleeping socks, with a high wool content.
Carefully get into the tent, bringing in the least water possible. Change out of the wet clothing, put on dry. If down is used (jacket, sleeping bag), only use this stuff in the tent/tarp — of course.
Don't try to dry things out. It's possible to dry things out at least to some degree, at the cost perhaps of getting more moisture into your sleeping bag/quilt. So I don't do it. Nor do I carry N changes of clothing for an N-day trip — this really doesn't work for trips of any length. Each morning just pack up most of your gear, then change back into those sodden wet clothes and shoes from the day before, then quickly finish packing up and start hiking. As Piper said, hike hard initially if you're cold. This is less fun when there's snow on the ground; not a big deal in warmer wet conditions. But once you're moving, you can tune light layers to stay warm enough. Set your pace to something sustainable, so that you don't need long breaks — it gets too cold to take long breaks, and IMO it's a PITA to fully pitch camp and change clothes except at the end of the day, plus the more times you do that, the more moisture inevitably works it's way into those things you're trying hard to keep dry.
This is certainly not as much fun as hiking on a sunny day, but it's very do-able, with some practice/experience and … very important IMO … that shift in expectations and attitude. Just enjoy being able to keep going in tougher conditions and look forward to that day(s) in future when it will warm up. Or at least to getting to your car or to a trail town motel room to dry out during a day off.
If the rain does stop and the sun comes out, definitely take an extended break in the sun to eat and spread things out to dry. Long distance hikers often have some clothing item(s) drying on top of the pack as they hike.
One of the tough things in the PNW is that we can get a rainy day followed by several days where it doesn't rain, but cloud cover plus temp/humidity is such that the vegetation just never dries out. So unless you're on a relatively wide and recently maintained trail, you can find yourself day after day rewetting a lot of your clothing from the brush. Shoes in particular can seem to just never dry out.
So what. Shift expectations and be happy.
If you're in really continuous rain, it's not a bad idea to eat cold, unless your tent is such that you can safely cook under a tent flap. A pee bottle is helpful, for males at least. Perhaps the new dry down sleeping bags/quilts are worth it (not sure) for cold/humid conditions. I.e., I'm less concerned about getting my bag soaking wet as I am about just using the bag night after night in high humidity, and inevitably losing some loft thereby over time. Hasn't been a big problem for me, however, with regular down.