I have an old Maytag dryer, model number DE303.  I would not be surprised if it's from the early 90s. The heating element stays on if the cycle selector is set to anything other than "Off" even if the door is open.  It also doesn't shut itself off during it's cycle, so I've taken to setting a timer on my phone or stove so I don't forget about it

I understand that finding parts for it is unlikely given its age, but I want to at least try to fix it before just tossing it. What would be the cause of this kind of problem?

I have a DE313 purchased in 1989 and have not had any trouble finding parts. Just have to search on the internet. There are several websites that will have parts.

These guys have a lot of parts: https://www.repairclinic.com/

They also have videos that may help you figure out what's wrong and how to fix it.

I've googled model numbers, found exploded diagrams with parts numbers, and then googled the part number.

I've found parts on ebay and amazon once I knew a part number.

Good hunting.

Link to eReplacementParts.com

Then again, repairing a 30 year old dryer may not be economical.

Consider a replacement unit, even if purchased used.

Looking at the diagram if everything is correct as it came new, you would have to have multiple parts bad for that to happen. I'd bet you have some bad wiring either not connected properly or one touching somewhere it shouldn't.

https://www.appliancepartspros.com/parts-for-maytag-de303.html

Once you find the part numbers you need, check Ebay and Amazon and Google.  I'm seeing the heating element on ebay for $50 and less.