Doesn't work. Känner is a transitive verb, meaning it requires an object. You can say jag känner trötthet ("I feel tiredness"), but it doesn't sound very natural.
How do you tell if something is a transitive verb? It would be immensely helpful to know this - Is it just a 'learning' thing?
I know that "mår" also translates to "feel", so I was wondering if "Jag mår trött" works, or is "mår" only used for other things?
No, with mår you're telling how you feel rather than what. I know that's not super intuitive, but it means you can må bra, må dåligt, må illa, må konstigt, etc. - but not må trött, må piggt, etc.
Could 'I know I'm tired' be used here? Seems like a more logical translation.
No, "Känner" only means "know" when you talk about knowing people.
Otherwise "känner" means "feel".
I know I'm tired - Jag vet att jag är trött
What is the difference between känner and känns?
Am I correct in thinking that känns refers to how something other than oneself feels, while känner refers to how one oneself feels?
Tusen tack! (My Norwegian friends say that—I don't know if it translates well into Swedish.)