DoctorHealth is by far better - 21 tld taken, against 2 for DoctorDrug. It also sounds much better. DoctorDrug has a negative connotation to me.
Now, both names are not actively on sale. So, you have no idea whether or not the owner will sell it, correct?
I'd start with 100$ then, playing that it's a ''side'', a non-important thing to you, but you could've used it for redirection or other things... And then it's totally up to him - to refuse, agree or go much higher in price.