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All animal manure should be properly composted prior to using on your garden.
These articles will help you.

https://www.gardeningknowhow.com/composting/manures/the-benefits-of-manure-in-your-garden.htm

https://www.gardeningknowhow.com/edible/fruits/watermelon/fertilizing-watermelon-plants.htm

Watermelons need full sun, a lot of water and a lot of room. Plant when the soil temperature at 4 inches is between 60 and 65 degrees F (15.6 to 18.3 C.) Provide compost rich soil when you plant and apply fertilizer according to the article below.

You can build a fence around your garden to keep out the rodents and wild animals. Build the fence 3 feet high with an additional 6 inches underground.

Please let us know if you have any other gardening questions and happy gardening!

https://www.gardeningknowhow.com/edible/fruits/watermelon/growing-watermelon.htm

https://www.gardeningknowhow.com/edible/fruits/watermelon/fertilizing-watermelon-plants.htm

https://www.gardeningknowhow.com/edible/fruits/watermelon/watering-watermelon.htm

This is pretty typical. Sometimes, new hybrids can be coaxed into producing more than that, but keeping seeds will lead to less than ideal progeny.

It will help you to start with fresh seed from a reputable breeder. This will ensure a better crop, of course, with proper care.

Here is an article to help you grow watermelon:

https://www.gardeningknowhow.com/edible/fruits/watermelon/growing-watermelon.htm

Is this the article you read below? Grow only one vine per pot. You can start with three or four seeds to make sure one grows, but if they all grow into seedlings, thin out all but the healthiest looking one. The article says to use at least a 5 gallon pot per vine. I would buy a packet of seed from a reputable seed company in one of the varieties recommended in the article for compactness and small fruit. By planting seeds from a grocery store melon you don't know what variety it is.

https://www.gardeningknowhow.com/edible/fruits/watermelon/growing-watermelons-containers.htm

Here is more information:

https://www.gardeningknowhow.com/edible/fruits/watermelon/cutting-watermelon-vines.htm

This appears to be multiple infections from heavy, oversaturated soils.

Fungicides and Bactericides will be in order, here. These articles will help:

https://www.gardeningknowhow.com/garden-how-to/info/using-fungicides-in-garden.htm

https://www.gardeningknowhow.com/garden-how-to/info/applying-bactericide-to-plants.htm

I would recommend adding several amendments to increase porosity of soil, drainage, and nutrient content. This will help to stave off infections in the future, once you have this one under control.

Here are some articles that will help:

https://www.gardeningknowhow.com/special/organic/the-importance-of-soil-for-an-organic-garden.htm

https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/improving-garden-soil.htm

You probably need to examine the leaves on your plants to see if they're being affected by a fungus or mildew.https://www.gardeningknowhow.com/edible/fruits/watermelon/watermelon-disease-control.htm

Here's another article that discusses watermelons that split: https://www.gardeningknowhow.com/edible/fruits/watermelon/watermelon-splitting.htm

There's no way to tell from here what varieties you've planted, but if you're finding ripened melons, regardless of their size, it would probably be best to go ahead and harvest them before they become over-ripe.