Will you still be a dependent for 2017 with this increase in income?

Full time or part time is irrelevant. If you claim 0 on the W-4, more income tax will be withheld from each paycheck. That will reduce the probability of you owing more taxes at tax time, or can increase the amount refunded to you at tax time if the withholding is to much because you claimed zero on the W-4.

Are you a full time student and how old are you?  If you make more than 4,050 in 2017 you might not be a dependent anymore.

The answer is really just to fill out the worksheet on the W-4 (front and back) and figure it out for yourself.  It has a table to calculate the impact of having two jobs.  You should definitely expect to claim single with 0 allowances at both jobs, the question will be, is that enough, or do you need to have a few dollars extra taken out.  The worksheet will answer that for you.

Separately, if you are 19 or older, and if you are not in school full time, you can't be claimed as a dependent if you earn more than $4050.  So you might claim single with 1 allowance on one job and 0 on the part time job.  The W-4 worksheet will take that into account as well if you answer that you can't be claimed as a dependent.