Can I apply any college without getting cutoff in jee main?

There are very limited college who provide admission on the basis of class 12th.
Some of the colleges are Amity, Sastra University and LPU.
To take admission in top colleges of engineering you should have cracked JEE Main Exam. Through JEE Main you can take admission in NITs/IITs/IIITs and GFTIs. There are many top private colleges who consider JEE Main marks to provide admission.
You can check the colleges below-
PSG College of Technology, Coimbatore
SSN College of Engineering, Kalavakkam
Thapar University, Patiala
Bharati Vidyapeeth Deemed University College of Engineering, Pune
Birla Institute of Technology, Mesra
Apart, from JEE you can also give below private level entrance exam as they provide very good placements.
1- BITSAT
2- VIT
3- SRM
4- Manipal
5- Amrita
6- COMEDK
If you want to take admission in government colleges you can give your state level exam.
You should give as many as entrance exam. So, that you have good options in your hand. Always prefer a college with good placements and faculty.