This part is inevitable in students preparing for NEET. And let me tell you, it only amplifies during MBBS, yet you learn how to manage the stress and pressure and get your work done.
Focus on preparing for your entrance exam. Sit down with your parents and discuss with them how you can balance Allen and college. Take their help.
I would suggest making a timetable for each week, so you know how much to cover without overwhelming yourself. Since NEET doesn't have any weightage to board marks, you can focus on passing with respectable marks. Also, look at NEET as an extended version of college syllabus, as the basic material for studying is same. By not compartmentalising Entrance exam and boards into different study groups, you take away the stress you are feeling.
On a personal note, do communicate with your parents, teachers, sensible peers when you are tensed. Talking about your fears will help you and people around you by letting them do their part. I would suggest that you talk to a counselor regarding your workload and really do take up studying with a timetable. Do try and incorporate some breathing exercises in your routine to freshen up and keep your panic at bay.
Most of the times the situation is much simpler than our mind conjures it to be. Talk about your issues at home and let your parents help you in the way adults can. All of us go through this phase where we feel we are too late and can't sort it out, but you do have time. Take action while you can and stay calm.