Your best bet for the Carrot is Bloodstone, in my experience. The food salesman is up by the furinture store. His max is 2 star (besides some fish), which is what the puny carrot is. It may take a few "sleep for 7 days" tries, but that's where I got the carrots for 2 of my heros & it wasn't very difficult at all.

Dye, check Fairfax Garden for clothing salesmen. If there isn't one - sleep for 7 days & check back. Do this for a potion trader as well. Fairfax is the best bet for 5 star items bar none.

Another trick I read in a different thread: if there is a clothing salesman in Fairfax, but they don't have the Dye - have them follow you, go sleep for 7 days, when you wake up they should still be with you - check again. Repeat as needed.

Travel between the 2 towns and check each time you sleep so as to not waste too much time.

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"I love things with sea shells and sea horses on 'em
Like blankets, and towels, and little bags
I love 'em! Sea horses. Forever!"