Is it overly formal to say 南开大学 (Nánkāi Dàxué) "Nankai University" instead of its abbreviation 南大 to people at 南开大学?
Unlike 中华人民共和国, the name of “南开大学” does not carry that level of formalness. It is alright to use it in conversation.

And the short form of "南开大学" is usually "南开", at least that is the one I always heard. You can use "南开" to make it more casual. This is what we often use to refer to "南开大学" in conversation.

(The wikipedia page of 南开大学 has such a note:


  《南开大学章程》中明确：南开大学的简称为“南开”。在民国时期，南开大学曾简称“南大”。目前，在天津等地区南开大学仍时常被简称为“南大”。


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And by the way, in casual talk, I would use "老师" instead of "教师"