My engine keeps shaking. The RPMs go up and down. Sometimes it even stalls.

how many miles on engine? plugs ever been replaced? computer is trying to compensate for engine miss, this will effect fuel trim and (i a c ) idle air control valve. rpm will go up and down, check EGR operation, and ports for clogging and if who ever pulls plugs make sure they have done a triton engine before. loosen 1/4 turn engine hot, soak plug holes with brake cleaner and PB Blaster penetrant let set over night then remove plugs. blow cylinders out meaning spin engine over before installing new plugs. hope this helps

There is no "One" answer for this problem. You will need to run tests to confirm. What have you checked, tested, repaired, or replaced so far? Is the Engine Light on? If so, I need to know what the codes are. Please have them read. From you description, it sounds like your vehicle has a vacuum leak. The ECU is having a hard time controlling the idle. Check for vacuum leaks at hoses, the plenum, and brake booster. Check the EGR and Idle Air control valve. Check fuel-line pressure (check it with the engine off and compare it to the engine running). It is hard to test O2 sensors unless you know how to but you can remove them and do the "Shake Test". All you do is remove the O2 sensor and shake it next to your ear. If you hear a rattle noise, it is junk. It is possible that one or more fuel injectors are dirty, clogged, or worn (causing this issue). The more information you can provide me the better I can help you narrow this down, good luck.

is the check engine light on? If so, have an auto parts store retrieve the codes.

The F 150 depending on the engine has a MAP sensor if the sensor vacuum get disconnected or even if the sensor fails the result is black smoke.RPM jumnping up and down look for a vaccum hose that is disconnected or broken.

it sounds like its drawing air into the inlet mannifold

either the fuel filter is blocked or the pump is weak ,

sounds like the idle air control is either clogged with carbon or it is bad. take iac off and clean out the passage way to engine. do not put valve in any liquid, only the end of valve can be cleaned were valve is. do not push on valve while cleaning or the adjustment will be messed up.depending on the size of engine it should idle at about 650 to 700 rpms.I have a 3.1 and it idles at 650 rpms,

Change fuel filter to start with. It sounds like it could be partly clogged. If no change, change plugs and wires. I'm not sure what you mean by the "rumble strip" thing, unless it's shaking as a result of one cylinder not firing, and causing a shaking feeling. A dead plug or wire would also cause loss of power and stalling. Let me know what you find.

try replacing throttle position sensor

Sounds like a bad motor mount, pop the hood and see if the motor shakes or moves rapidly when u put it in gear holding down the brake.