I think you can overcome this disposition, once you realize listening to the music has absolutely no power whatsoever to invade your personal beliefs. One can absolutely hold respect for the product of an institution (now) foreign to them and bask in its splendor without accepting its message as dogmatic truth.

Ask yourself honestly - what is it that really causes such aversion? If it's the perception of others, it shouldn't be. Why deny yourself of something that moves you "to tears" based on what others think of you? You need to feel comfortable in your own skin, and feel confident in your beliefs - whatever they may or may not be, and know that in no way does appreciating this music change who you are, or your perspective or outlook in life.

There is also much secular choral music you can enjoy that is just as stunning as liturgical and sacred music. One doesn't have to be an atheist to enjoy them, just as one doesn't have to be religious to enjoy theological work. Try shifting your focus toward appreciating the music from a historical aspect - or simply just based on grandeur and beauty of the music itself.

For the record, no - you are not the first person to feel this way - not always, but quite often, the issue people (who have shared a similar background) have is not with the music at all - it's with finding oneself, and being comfortable with who they are now - and realizing the past doesn't define who they are today. The only one in control of your personal beliefs is you. Don’t be so hard on yourself - enjoy the music! :)