I do laundry while camping often and clothespegs truly are a good solution to the problem of securing laundry. That said, you have some options.
First, consider using the clothes to secure themselves. For example, unbutton a pair of pants, then close the button up around the line so it hangs. It won't dry as fast as if it was pegged out, but it will be faster than if it was lying on the ground after falling off. Wrap a shirt around the line horizontally - neck end to your left, say and waist end to your right - then do up the buttons to hold the shirt on. Or undo one end of the line and put it through a sleeve, and out the neck, or through a sleeve, across the body of the shirt and out the other sleeve (works for Tshirts as well as button shirts.) These things won't help you with your "smalls" but might let you conserve a limited number of pegs to use for those things that really need them.
Or, you can buy a pegless washing line. This is made of two or more stretchy lines wound together. You pull the two apart a bit to stick a corner of a towel or clothing item through the hole, and when you let go, the elastic holds on to the item.
Many (but not all) are marked Indoor, but you could try them outside. There are also pegless lines made of plastic clips that are marked outdoor.
I found an elastic version on Amazon for less than $3, though you might need more than one to get enough length for a full load of washing.