While some schools may have a biomedically related biomolecular concentration under ChE, I would say that is rare, mostly biomolecular engineering is geared towards production of biological chemicals. If pharmaceuticals fulfills your desires for biomed, that would be perfect. Otherwise, I would only specialize in ChE if they have a specific biomedical track that matches your interest (tissue engineering could possibly be there …).
Tissue engineering may be covered in material science.
Prostheses like limbs, are mechanical, but there are a growing number of electrical devices, monitors, etc that could involve EE, especially related to heart and brain which have a lot of electrical activity.
Since you have 3 schools in TX that you are considering, I would spend some time with all 3 websites looking where they have bio-medical related coursework, what they specialize in both coursework wise and research wise, and how it relates to your interests and abilities. Since you have all your core coursework done, you are stepping right into junior year classes on their 4 year plan, so that list and the coursework should either excite you or bore you.
Also, if you can manage, a scheduled visit seems like a great idea, could be an open house or a grad student tour or more.
The four specialties in biomed relate to EE,ME,MatSci, and CS in most schools.
All engineering specialties offer the possibility of helping society deal with technological issues and improve people’s lives. Clean water is essential for example.
If you get a BME degree and can’t find really technical employment, you could pursue a MS at night with employer funding or pursue a MS on your own funding. High tech design in many fields requires a masters, R&D money is limited and there are a lot of MS and PhD people who would love to have an R&D position in any field.