This is definitely a TurboTax bug; someone from Intuit helped me out in a different thread. I had to dig into the TurboTax bowels and delete my 1099-SA data and start over, and then the New button started working.
https://ttlc.intuit.com/community/taxes/discussion/re-how-can-i-add-more-than-one-1099-sa-the-add-bu...
