1. Haven't you gone to the market yet? We need more mushrooms and carrots! = 아직 시장으로 가지 않았어요? 우리는 버섯하고 당근 더 필요해요!
=> 아직 시장에 안 갔다왔어요? / 안 갔어요? (우리) 버섯하고 당근이 (더) 필요해요! / 있어야 돼요!.
To get something from a store, we say (X-를 사러) 가게에 갔다오다 or 다녀오다(simple 가다 is also used sometimes as a short form). 으로 is never used in this context, since 으로 indicates your destination, not a temporary stop. 에 is the right particle here.
2. I have to drink more water. = 저는 물을 더 마셔야 해요.
=> 저는 물(을) (좀) 더 마셔야겠어요.
마셔야 해요 sounds a little too heavy for such s small thing. If you just want more water, using the lighter ending -야겠다 with the adverb 좀 (it often lightens the meaning) is more natural.
3. The weather is really cold so I will have to wear more clothes. = 날씨가 정말 추워요, 그래서 저는 옷을 더 입어야 할 거예요.
=> 날씨가 정말 추워서 옷을 더 (끼어) 입어야겠어요 / 입어야 할 거 같아요.
Again, -야겠어요 (same as -야 하겠어요) is good for "I should ...". -야 할 거예요 is also good, but this phrase sometimes doesn't work well for the first person "I", because it sounds like predicting an obvious thing about oneself.
끼어 입다 is a common expression for wearing something on top of other clothes as in this example. 끼어 입다 or 껴 입다 is typical when asking someone to wear more warmer clothes for the cold.