There are basically two possibilities here:
You're already legally Hungarian, but do not yet have the documents to prove it. In this case, you should be able to stay in Hungary until you get the documents. (If you want to travel to other Schengen countries, you could in theory do that with the same evidence you're using to document your Hungarian citizenship, but without a passport or ID card it's not certain that other authorities would accept you as an EU citizen, so it would be best to avoid this.)
You're not yet legally Hungarian, so still subject to the 90/180 rule. In this case, as a pre-doctoral intern, you may be able to qualify for a D visa or residence permit in the country where you're doing the internship. If so, that will help you stay in the Schengen area. If you're not able to get a D visa or permit, there's no legal way for you to stay in the Schengen area.
Your lawyer should be able to tell you which possibility applies to you. Your lawyer can also discuss with you the consequences of overstaying if the second possibility applies to you. For example, if you're certain to receive a Hungarian passport only days after you're supposed to leave the Schengen area, it might make sense to stay anyway, or it might make sense to book a short stay in Croatia or another non-Schengen jurisdiction so you can remain within the 90/180 restriction when you return to pick up your passport. If the delay is indeterminate or is known to be long, then it might be better just to leave if a D visa or residence permit is not possible.