Rather than being scared of deadlines, try to value your own work and want to get the best grade you can. Don't you want your work to be graded on its merit instead of losing points just for lateness? Time management will really help you here. Starting things earlier and imagining the deadline is a few days before the actual deadline will give you those couple of extra days to finish without getting points deducted for lateness.
Would your professors be open if you approached them about getting some extensions? I always offered to extend deadlines if a student reached out to me at least a week before something was due, if they knew they were overwhelmed and needed more time to do a good job.