Hey,
See, 60 is not too good score in JEE Mains but if you compare your score with others in your city it may be approximately same. With your score, you can surely get an engineering college. Don't expect NITs or other government colleges, but you can get a decent college in your own city or maybe of your state. It's all relative so you don't need to worry much. First, fill JoSAA application form and also the form of your state counselling. There are also some private colleges which offer admission through JEE Mains. You should also try some of the open private colleges which take the open test like VIT ,SRM, BITS and UPES. You may also try for open universities like DU which offers admission on the basis of your board's score. Don't feel disheartened, you have so many options and also you are having a decent score in Mains, you should surely get a good college.
All the best.