What do truckers do after they hit the 11 hour mark for driving?

You've asked two important questions that new drivers have. I'll respond with some strategies for maximizing your miles/income and reducing stress.

First, as others have already noted, you DON'T have to drive all 11 hours AND once you are done driving there's nothing that requires you to continue working or staying on duty OTHER than doing a post trip inspection.

If you are on a multi day load, then you should "work up" on your schedule. By Work Up I mean on a day when you have nothing to do but DRIVE (no loading or unloading) you should be able to drive 11 hours within 12 hours. You will have about an hour worth of pretrip, post trip, fuel stops, and mandatory 30 minute breaks. Let's say you started at 6:00 am. You finish your post trip inspection at 6:00 pm. You then immediately log off duty or sleeper berth. 10 hours later you can go back On Duty for your pretrip inspection at 4:00 am. Now you have "worked up" two hours. Do that again and the next day you start at 2:00 am.

Don't dawdle and try to start every day at the same time. TCB (Take Care of Business) and get to your delivery as early as possible. Contact the receiver to see if they will take you early. Communicate with dispatch and let them know when you expect to arrive so they can start planning your next load.

Second, you asked about what if the truck stop is full. Now you are out of driving time and you are screwed. You can sit in the middle and wait for another driver to vacate a spot or "make a parking spot" where there is none, fully expecting to have someone bang on your door because you are blocking their way (and then hopefully snag that empty space). But there is a better way.

Plan ahead. Sometimes I KNOW parking is full hours before I can get to a place that maximizes my hours of driving. That's the time to plan to cut the day short, say 8-9 hours of driving, in order to get a safe place to park.

Download the Trucker's Path app. You can look ahead to see if a parking lot is full or has plenty of parking. Consider taking a 10 in places other than a truck stop or rest area. About two hours before I plan to end my day I take a break and do some research. I use Google Maps with Satellite View to see if there are WIDE areas beside the road, or a parking lot with wide, truck friendly entrances (Hint: not all Walmarts are truck friendly).

Have a plan where you will park. Have a backup plan and a backup plan for the backup plan.