Sounds like you might need a little more room. First, it is usually suggested that no more than 3 plants per hill. These vines will run like crazy, possibly up to 10 feet. You can trellis them, but it is a lot of work, because the melons are heavy and need support. 10 Sugar Baby vines is a lot to deal with. I would probably thin them to 2 of your healthiest plants and go from there. They should grow fine in basic garden soil with a little fertilizer.

Watermelons produce about one lb per sq ft. So your area is big enough to produce one small melon. That only takes one plant. Either that or they need more room, out or up. My watermelon area is about 30ft by 30 ft. If I can keep them producing from July to October I might get 1,000 to 1,500 lbs. To produce that long takes careful watering, fertilizer, and spraying for foliar disease.

What it really boils down to is that your area isn't big enough for watermelon. If you can't let them spread out they need to go up, as sated above.

Can I allow them to sprawl out into the grass around them and train them to grow that way? Here is the basic layout of my garden. http://imgur.com/5OFsu Also, what types of fertilizer should I use?

They can grow on the grass. It's not ideal either for the grass or the melons. I prefer growing them on black plastic or weed barrier. But in your case that might kill the grass. Add some high nitrogen fertilizer if needed.

So I should just cut 2 square holes in the fencing and allow them to sprawl out on the grass? How big should the holes be?(On a side note, if you copy and paste the link, I recommend clicking the image again to enlarge it. There is some pretty small text I put there.)

If the chicken wire has holes about 1" in diameter, the vines can just grow through the wire and will be fine.

I would think you have enough root room for the plants, it is just there isn't enough room for the vines if they stay confined in the raised bed so will need to sprawl out.