"몸조심하고 아프지말고!! 제발 아프지말고ㅠㅠ 몸 조심히 재밌게 잘 즐기고와. 날씨 추우니까 옷 따뜻하게 입고 밥도 잘 챙겨먹고 알겠지요? 화이팅!!" There is no serious error, but if it was an exam, it might get some negative points. To be correct in orthography, it might be written in this way: 몸조심하고 아프지 말고! 제발 아프지 말고, 몸 조심히 재밌게 잘 즐기고 와. 그리고 날씨 추우니까 옷 따뜻하게 입고 밥도 잘 챙겨 먹고 알겠지요? 화이팅! ('그리고' added) For "in addition" before 날씨 추우니까, I think 그리고 is the best choice. 게다가 comes from 거기에(there)다가(adding to it), so it's like "on top of it" or "furthermore", with a feel of logic. In this cases, it's not making a further point but just casually adding something, so 게다가 sounds strange. Here's a 게다가 example: - 그 회사는 작년에 큰 적자를 냈어. 게다가 주식이 상장 취소됐대. => That company incurred a huge loss last year. In addition, they say its stock got delisted from the exchange.

that's 100% correct "게다가" is not suitable. "그리고" or "또," would be better or "덧붙여서" can be used

Hey, you did good job! The sentence sounds perfect and you don't really need to put "and" there. but if you want to make it perfect, you can put "그리고". For the conversation between friends, we are not using "게다가" that much, and I'm not sure you use "알겠지요?" on purpose because that is the only formal expression you used in this sentence. If you wrote it to friend, it should be 알겠지? instead of 알겠지요?