No you cannot. "I need to make an appointment, but I also need to check my schedule because it is always full." "I need to make an appointment, but my schedule is always full." "I need to make an appointment, but I also need to check my schedule because it is always busy." "I need to make an appointment, but I am always busy." "I need to make an appointment, but I also need to clear my schedule." "I need to make an appointment, but I also need to clear my schedule because I am always busy." And you can swap and interchange these parts; check/clear -- busy/full in the different sentences.

No, sorry - that doesn't make much sense. 'I need to make an appointment' is fine, but it isn't clear what you're trying to say in the rest of the sentence. The phrase 'on a narrow' doesn't mean anything at all. 'Narrow' is an adjective: it's the opposite of 'wide'. There is no such thing as 'a narrow'.

If you want to use a phrase to describe your schedule as busy, you could say: "My schedule is always jam packed".