Maybe to wet underneath, too much rain, etc.
I will put some straw under them so they will be dry.
Also, they prefer full sun.

That would be my question - what causes them to rot? It isn't as if it is a normal part of growing them. So is it ground contact? Thick mulch fixes that. Is it pests or disease? Poorly draining soil? Not enough sun? Left too long on the vine? Planted too early?

If you can ID the cause then there are ways to fix it.

Dave

I just re-read your question.
You said that the fruits rot at EARLY STAGE: ahha!
Melons, cukes, sqash, etc have two kinds of flowers;
male and female. The female has a tiny fruit at the base.
If the female flower does not get pollinated, that tiny fruit might grow, say as big as an egg, walnut. But it will rot or
it will be aborted. If that is what has happened to you, then
you probably have pollination problem. Honey Bees and bumble bees are the best pollinators. If they dont come around when the flowers open, then you will not have wM.
Another possibility is that they DO come around but there are not male flowers, or very few of them.
The solution to this is that you have to do the job and
hand polinate them yourself. You have to do this during the first few hours that female lowers open. Pick a male (be careful not to shake it too much). Bring it over a female and give them a gentle kiss. Move on to second or third.

Good catch Cyrus! If it is either inadequate pollination or BER (blossom end rot) then both of those problems can be identified and corrected easily.

How to pollinate by hand FAQ

What is BER and how do I prevent it FAQ

Dave

Thanks everyone, I think you may have solved this one for me.

For the larger watermelons a person may think that 5 or 6 melons at the first setting would be better than two...think again. Quality will suffer.

I planted a watermelon plant for the first time, not really knowing much about their growth and I was really taken by surprise at how quickly they grow and spread. My husband freaked out about the plant taking over the entire yard and asked me to remove the plant. I couldn't do it. There were little fruits already sprouting up and I just didn't want to kill it entirely, so I have been trimming back the vines. Will this cause damage to the plant or affect fruiting?

Usually you can keep turning the vine tips to keep things more compact. Heavy trimming likely will hurt quality as will too many fruits on the plant.