No you don't need other certificate but you have less Chance to get admission in Delhi because they have only 15% reserved seats for people of other state and if you'll get a certificate of Delhi government then you'll fall under 85% people to get admission

Dear aspirant

You have done your schooling from Delhi i.e,10th and 12th then by default you are eligible for Delhi state quota counseling but if you want to have benifits of category reservation also then you will have to provide the caste certificate from that particular state i.e Delhi otherwise you will be not eligible for category seat instead of that you will be Gen category candidates in Delhi state counseling.

As you have caste certificate from up and if you have an domicile of up state then you can participate the counseling of up state counseling and can get benifits of your category also.

For any state counseling you should have domicile certificate and caste certificate from the same state for which you are applying the counseling to get full benifits of your category reservation.

AS you are in last days of neet preparation,so for time being now try to focus on scoring good marks by revising your concepts and formula and once exam get over you can think about all these things.

hope this information helps you

all the best