Hello nicerhugs1286,
about your question between 'jeo-neun apayo'? and "Je-ga mani apayo"? Between je-ga vs. jeo-neun
First, I am sorry for you "I am sick today".
Hope my explanation makes you feel better...
Second, about 저는 아파요 (jeo-neun apayo) is totally fine (in my opinion).
However, 제가 많이 아파요 (je-ga man-i apayo) sounds more... like 'pitiable'.
Third, about "I".
"I" can be translated as 나 (na) in INFORMAL Korean and 저 (jeo) in FORMAL Korean.
so, it becomes... 나는 (na-neun) or 저는 (jeo-neun).
But, here is one more thing to know about... when you add 'POSSESSIVE PRONOUN' - 의 (ui) to them, they become...like
나 (na) + 의 (ui) = 내 (nae) in INFORMAL, which literally means "my". For example, you would say "내 컴퓨터" (nae keom-pyu-teo) "my computer" to your younger sibling or close Korean friends.
저 (jeo) + 의 (ui) = 제 (je) in FORMAL, which literally means "my" as well. For example, you would say "제 컴퓨터" (je keom-pyu-teo) "my computer" to your elder sibling or parents or parents or your boss at the office.
Last, these - 내(nae) and 제(je) - can be used for the SUBJECT as well.
For examples, 나는 아파 (na-neun apa) "I am sick" = 내가 아파 (nae-ga apa) "I am sick" (when talking to your K-friends)
저는/저가 많이 아파요 (jeo-ga man-i apayo) "I am very sick" = 제가 많이 아파요 (je-ga man-i apayo) "I am very sick" (when talking to your boss).
Yes, the most polite expression would be "제가 많이 아파요".
That's why your co-worker recommended you to say like that.
Remember this - 제 (je) is "my" in FORMAL Korean but, it can be used for the SUBJECT "I" in FORMAL Korean.
Hope my explanation helps you understand Korean.
cheers,
Tim