If I plug them back up again it trips. What's wrong?
Sounds to me like a grounded element. You cannot always find the short visually re: a cracked insulator etc.
Check it with a meter.
Unplug the unit and remove both wires from the heating element.
Measure the element usually around 10 ohms.
Then check each heater connector to the frame, both should be infinite ohms.

If you do not own a meter, I would suggest you purchase a one. You can get a decent digital multimeter for under $20.00. You do not need fancy though it is nice if the leads are a couple feet long.
If it saves ordering one unnecessary part it has paid for itself and you end up owning a useful tool.
Most places will not let you return electrical parts so if you order it, you own it.
A couple things to watch when measuring ohms and continuity
1. Always remove power from the machine otherwise you could blow your meter.
2. Always disconnect at least one side of any device you are checking. This eliminates the possibility of measuring an alternate/parallel circuit path.
3. When checking for closed contacts and continuity use the lowest scale (Usually 200 ohms). Then try higher scales. This scale is 0 to 200 ohms so if the device you are measuring is 300 ohms this scale would show an open circuit which it is not, you are just measuring outside the scale's dynamic range.

There is a good STICKY at the start of this forum about it's use.

NOTE: I use mine constantly around the house and in the garage.

Usually one side of the heater goes to a centrifugal switch on the motor and then to one side of the line. This switch closes when the motor is rotating and provides the heater with power. It is there to ensure that the heater does not get hot without air blowing over it.

The other side goes through the thermostats and a set of timer contacts to the other side of the line. Completing the 240 volt circuit.

I cannot really say what is going on here.
Note if the centrifugal switch contacts are welded together or the mechanism on the motor which closes the switch is not resetting, the heater will get power whenever the unit is plugged in other than at Off. But this should not blow your breaker.
It will regulate the temperature using the hi-limit thermostat, till this thermostat's contacts weld together and then it will blow the thermal fuse.

This has to be what's grounding it though right?

Yes it does sound like the element is grounded to the heater case.

Do I need a new heater case?
I could not find the parts for your unit using the given model number. I looks like it is short at least 3 numbers. But on most units you would need a new heater assembly.

Why would it only do it when angled up into the dryer?
Because when it touches the machine frame it completes the circuit as the frame is grounded back to the electrical panel.
When hanging loose it is floating and the case becomes live but there is no where for the current to go.