OK, I'm really not supposed to give advice without seeing the patient, but I'm going to make some general comments. Please take this in the spirit in which it was intended, that I am not making a specific recommendation for the OP, whose ear status I do not know. If you need some drawings to follow along with the anatomy talk, look at
this page.
1) After a radical mastoidectomy, the middle ear space is obliterated, leaving a large "bowl", which is physiologically similar to a normal ear canal. Since there is no air-filled middle ear space, the usual issues of equalization do not apply. However, there are all sorts of variants of mastoidectomy, and in some cases there is still air behind the residual or reconstructed eardrum.
2) It's generally a good idea to keep a mastoid cavity dry, but there are certainly people who swim, shower or otherwise get the cavity wet. Whether this would cause a minor annoyance or a major case of otitis externa (swimmer's ear) depends on the patient's particular anatomy, so if the OP hasn't seen an otologist in a while, that might be a good question to ask. Also ask if there are ways of drying the ear canal after water gets if there is an earplug failure. Depending on the OPs anatomy, it might not be the end of the world and otits externa might be preventable after exposure by using things like ear dryers (little hair dryers for the ear), certain drops, etc...
3) Diving with a tight fitting earplug will create an unventilated space and set the diver up for a squeeze. Ventilated earplugs exist, and some divers wear them, but it would be unlikely that you could get one for a post-mastoid bowl type ear canal.
4) Fixing this problem with commercial dive gear, especially modding a surface supplied rig to work with double tanks involving an incredible amount of task loading and potential failure points. This does NOT sound like a reasonable solution for a brand new OW diver. I'm being polite. The dryhood or the pro-ear thing might be better choices than that (again, haven't used them myself).
5) I don't know much about full face masks, but I don't think that they cover the ears.
6) If you want to PM me with your location, I may be able to refer you to an otologist (ear specialist) in your area, if you haven't seen one in a while. Sometimes, it is worthwhile to do a surgical revision of a mastoid bowl if there have been a lot of problems with granulation tissue, bleeding, infection, etc... There is also an operation to obliterate part of this space which makes it easier to keep clean.
Hope this helps!
Mike