How can I be removed as a medical POA of my cousin?

Make the distinction between medical and financial/legal proxies. A medical proxy shouldn't require the kind of care you mention; it's generally for situations in which your cousin can't make the decisions herself, such as end of life decisions if she's in a coma.

You can exercise the duties in the POA without being a hands on caregiver as stated above. Normally, the POA makes decisions on the person's behalf. Or carefully read the document and see if an alternate is named. If you can't serve the alternate can, in most jurisdictions.

And if that is not possible, IF she is competent, she can appoint someone else in a new POA document. I'd see an attorney about that.

And if that is not possible, I'd consult with an attorney about the next option, which is Petitioning the court to appoint someone.

If it helps, being a POA is not an obligation to be a hands on caregiver, although many seem to think that is part of it. You can step back from being the go to person whenever she needs something and still retain that authority if you choose.
Wikipedia sums it up as a "written authorization to represent or act on another's behalf in private affairs, business, or some other legal matter" or, as a healthcare proxy when when "he or she is incapable of making and executing the healthcare decisions".

She can appoint someone else as PoA.