You'll be fine, just use your ear plugs.
Don't stress about bringing them out, just be straight up and say that you ears are messed up and that the plugs help you - too easy! I used to always second guess using my ear plus in certain situations (movies, bars, loud restaurants) because I didn't want to bring attention to myself, and my friends can be a pain in the ass, but now I just use them whenever I need and I've realized no one cares or even notices.
Volume levels between cinemas and movies do vary greatly though, so it's something to take into consideration.
Ideally no one should need to use ear plugs at a cinema, and some venues are indeed extremely professional and make every effort to keep the volume at a sensible level. However, other venues have no clue and basically blast the sound system as loud as they can because they think it will attract more customers or enhance the experience somehow.
Volume levels also vary greatly depending on the type of movie. Big budget action adventure flicks like to emphasize dynamics to create tension, so one section of the movie will be super quiet and then as soon as the action begins, they blast off with sound effects and a huge soundtrack. On the other hand you have indie movies that have no soundtrack, and the volume basically stays the same throughout the entire feature.
So taking those factors into consideration, it's just a good idea to have ear plugs handy. Custom plugs with filters are great in the scenario because they generally provide a better balance of sound and allow you choose the level of attenuation (eg. 9db vs. 15db vs. 25db of protection).
In regards to bone conduction, I don't think this is an issue in cinemas - even with loud movies. I think ear plugs are more than adequate protection in this situation.
I've never been in a movie where I thought the bass was loud enough to damage my hearing, this is before and after my experience with tinnitus and hyperacusis. In comparison to bars, clubs and concerts where there are sub woofer stacks blasting off in your face for hours on end, the exposure in cinemas is minimal.
I think most ear plugs would be effective at protecting you from any brief exposures to loud bass harmonics during a feature. Having said that, everyone does react differently and I wouldn't discredit anyone who might disagree with me or unfortunately experienced otherwise, but I think the majority of us are fine to go and see movies with some hearing protection handy.
Anyway, go have fun!