Either one of these would be a fine choice that can set you up for a bright future, but the HBCU experience is really unique and not easily replicated elsewhere. I went to Spelman College for undergrad, and given the choice to do it all over again I would in a heartbeat. There’s something intangible about a place explicitly made for you, both in the education itself and also in the social experience.
Howard is the number two producer of African American doctors in the U.S. (HBCUs and the Production of Doctors - PMC). I have a PhD in a health sciences field. Going to an HBCU had a strong influence on my choice of research area and concentration, but it also inflected something else: I know so many young black professionals with PhDs, MDs, DDS, and other doctoral and professional degrees.
It boosts your self-efficacy, your belief in your ability to succeed: the message that these careers are not for you is diminished because you have so many strong examples to the contrary. After I attended Spelman, I never had any doubt in my mind about whether I could finish my PhD if I wanted to, at least not because I was black. And I was connected with a network of others that I could lean on for support when I needed it, and vice versa. Of course it’s possible to form that kind of support network without going to an HBCU, but it certainly helps.
As for your social anxiety, going to an HBCU taught me that black people were far more diverse than I had ever imagined (even as a black person myself). I found several of my tribes there: the nerdy anime lovers who wanted to study abroad in Japan, the hard driving student-government types, the other nerd kids who wanted research careers…HBCUs are packed with kids who were in predominantly white schools most of their lives and wanted a different experience. (My freshman roommate was one of them.)
That said, Tulane is a fantastic school, and New Orleans is a pretty black city. Xavier and Dillard are not far away; hanging out on campus doesn’t replace the HBCU experience, but it may be enough for you depending on your preferences and choices. And black students can and do have wonderful experiences at PWIs all the time. It’s just a different environment, and it really depends on what you’re looking for in a college.