If your last frost just barely happened, it's probably still too cold for the plant to be growing very much, especially if you have an insulated soil type (like clay or clay loam). However, the temperatures you mentioned aren't terribly cold (I'd be surprised if it were that warm every day, soon after the last frost, though); but they're not as warm as is optimal, though. Just be patient. Your plant should be fine (if it doesn't get much colder very often). I do believe fertilizer could help.
If you can put some black plastic on the ground around it, that can warm up the soil. Just cut a hole in the plastic where the plant would be, and put it over. Even a black garbage bag repurposed can be effective. I'm using black plastic for my watermelons, tomatoes and such, this year, and it seems to make a difference. I can feel the heat coming from under the plastic.
I tried the garbage bag on some tomatoes that we didn't have enough black plastic for, and they perked up quite a bit (they had been without black plastic for a while).
Be sure to water under the plastic, too (not just at the base of the plant).
Transplanting too early can hurt watermelon plants, even if it doesn't frost, perhaps especially if the plants aren't very old. Also, don't transplant too deeply. In my experience, the plants grow faster and survive more if you just plant them almost at the same level as they were before. Just put a little soil over the top of your seed-starting media. You don't want soil getting on the foliage: it can stunt plants if you don't wash it off. Also, soil being too near the growth tips seems to slow growth.