"I am interested in what do you through hikers do?"
Well, I am not a through hiker, really. I only head out for a couple weeks at a time. Rain is one of those times you sort of wish you were elswhere...out of it. Hiking is not a problem, usually. Wet trails, mud puddles (6" deep), slipery roots and rocks, skidding on duff...that is a problem. It will slow you down as you know.
I am usually hot and sweaty, so, getting wet with rain is two fold, it makes you wetter, it makes you cooler.
Your shoes eventually wet out. Less than good socks will saw away at the skin leaving sores...very close to blisters, but not quite the same. Like walking on after a deeper stream crossing, you should stop and drain your shoes and wring out your socks. Even if it continues to rain. Logging through a wet trail is about the same. Depending on how wet your shoes get, stop periodically and wring stuff out.
But, hiking in the rain can be enjoyable. Most hikers don't like it so the trails are usually free. Stopping under a good dry tree will help as you survey the area. Generally you just keep slogging. Often, the first trail shelter you find is a good place to stop and take a break. By now everything is wet, including your rain gear. You might get cold. If you do, you have two choices, Continue on or Stay put and set up your camp. If everything is wet except for your sleeping gear, you need a fire of some sort. Difficult at best. But it is possible to gather enough smaller stuff and birch bark for a small fire under a tarp. For the next couple hours you sit there tending the fire. It WILL go out if you do not tend it. Again you need to decide if it will get a bit dryer. Stay and cook supper? Move on? Stopping always means at least an hour of dryer time under a tarp with a fire. Pick the dryest spot and set up your bed, cook supper (even if it is after 1500.) Rain is not fun but it won't melt you. I usually reserve the "stay" option to after 1500 or when my fingers are wrinkled from the rain with no let-up in sight. There is always tomorrow to hike...
Always keep you sleeping gear dry. I make a set of sandals out of my liners to use around camp. Drying your feet will help. I use quick drying clothing, but "quick" is relative. Leaving it on will dry it much faster, but, it will also make you much colder. Down goods (jackets, bag) do not fare well in rain. Keep them dry and never put them on over wet clothing (they will get wet too.) Shoes, socks, etc will NOT dry. Get them as dry as possible by tieing the laces over the soles and hanging them over a cloths line, heels/scree collar down. Wet shoes will drain for about an hour even if they do not dry. Rinse and wring your socks. Hang them with the toes up for best draining.
On a weekend or three day hike, you *can* put damp cloths on top of your bag as you sleep. Not recommended for longer, though. After two nights the loss of loft/insulation may mean cold nights.
Your food should be in a dry bag. For a couple days it will not spoil, but if it gets wet, you need to eat/cook it. It may survive another couple days after cooking but this means carrying the extra water weight.