Mystic is a English word borrowed from Latin. It refers to experiencing the mysteries and to spirit initiation. A person using this term to replace a prior use of shaman might like to append the term mystic with another descriptive also from Latin or English to round it out and be more specific. So many folks have added terms like Plant, Animal, Wolf, etc to shaman to clarify, it would be just as easy to say Wolf Mystic or Plant Mystic or Animal Mystic. Mystic does not refer to any religion, and the term can be applied to nearly any religion. You could say Buddhist Mystic, Animist Mystic, Christian Mystic, Pagan Mystic, or even Neo-Pagan Mystic.
Of the many options: magician, wizard, druid, witch, sorcorer/ess, etc this one has the most strength as a good replacement for shaman. Mystic is an non-gendered word, it is a classical and English word with clear linguistic roots relativity unchanged from classic times so it does not step on any endangered cultures. The baggage for this term is low, I can’t think of any really, and I am not sure why people aren’t using it more already.
*Replacement of the term is only one possible step in rooting out cultural appropriation in neo-pagan practices. Replacement of the term alone while continuing to appropriate indigenous practices would be half assed to say the least.