Below is my standard blurb on this type of problem.

Check the heating coil.
Unplug the unit and both wires to the coil.
Check it with a meter, should be around 10 to 12 ohms.
Then check from each side of the coil to the case/frame, both should be infinite ohms (open). If not the coil may have sagged or broken and is touching the case. This can cause it to run on high and the thermostats cannot regulate it so the thermal cut-off blows.

The hi-limit should have regulated the temperature so the fuse did not blow, that is why there is a new one with the thermal cut-off.
The problem is that it's contacts are not as robust as the main temperature thermostat so after a while it's contacts weld together or stick closed. The temperature then rise and this blows the thermal cut off.
Note: That unless there is another problem in the unit the hi-limit should never have to open. It is just a safety device with the fuse being a backup safety device.

Just in case it is not a grounded element.
With all the below the high limit will also have to be replaced.
Check that the belt is OK.
Check the seals (drum etc) in the unit. The air is pulled over the heating coils, through the drum and pushed out the exhaust. So any large seal leak will pull in room air and the cycling thermostat on the blower will run the unit hot.
Check that the lint filter is not coated with fabric softener residue which greatly reduces air flow.
Check/clean your vent system.
Check/clean the blower wheel.

If all OK you may want to replace the cycling thermostat as it's contacts may not be opening (welded shut)