There are a couple of things I could suggest; For a more discrete option, you could play low level music on a phone or Digital audio player with a pair of Inner ear monitors (IEM's) fitted with some good after-market silicone ear tips such as SpinFit. SpinFit tips come in different sizes so you get a perfect seal for your ear.

Another option is you could buy some highly efficient sealed back over the ear noise-cancelling headphones. Choosing an efficent pair should mean you can use them with a phone or DAP without the need for a portable headphone amp.

You might also want to see an Audiologist to have your hearing tested. They might be able to diagnose the cause of the sensitivity, or otherwise refer you to a specialist who can provide a diagnosis and options to address the issue. Good luck.