There is no law against CCing a firearms in a hospital.
As for the jail, en-route lock it in the drug box.
Agreed, there are no
laws regarding CC in hospitals.
However, there are local, state and national regulations, professional rules, and employer rules that get in the way of EMT/paramedics/first responders/firefighters etc. carrying on the job. Just as there are policies in hospitals about private citizens carrying OC or CC on the premises.
If you, personally, want to risk your employment status by carrying when faced with rules/regs/policies that say NO, then go ahead. But it seems a bit reckless to be encouraging someone else to risk their job and their future employment/employability. Losing national certification over an issue like carrying OTJ is likely to cause you to learn how to say "Do you want fries with that?" in several languages.
As for locking your firearm in the drug box before you go in the jail - how will that help if you are carrying the drug box in with you? Or if you end up transporting a jail inmate and have to access the box while enroute.
Sorry, Joe, but there are a fairly sizeable number of folks who have to make decisions every day about either standing flat-footed for their rights or earning a living.
stay safe.