Late to the thread, but hopefully you'll see this & maybe it'll work for you.
This is not really anthro/fur so much as sci fi/alien, but,
David Brin wrote two books, Startide Rising and The Uplift War. They were also published as single volume called Earthclan. His aliens are diverse and weird, and some qualify as fur/anthro. During the course of these stories, there are a lot of little details thrown in all over the place about how species use a vast array of communication methods, including body language & psi, an avian species that dances, etc. He gives platform to miscommunication errors as key plot points. I think these will give you value, even if it isn't usually first contact. Should those two books work for you, then move on to the Heaven's Reach trilogy, in which these communication issues get even more treatment, and the stories get pretty intensely bizarre.
Along similar lines, CJ Cherryh has written some intense sci fi around first contact and various communication methods/issues between species. Hestia, the Pride of Chanur, Serpent's Reach, 40,000 in Gehenna. The Chanur novels are just head candy fun for /hmofa/, the other stuff is a bit more exotic, esp. Serpent's Reach (scent/pheromone communication with insectoids) and 40,000 in Gehenna (non-verbal integration with reptile society). Voyager in Night covers an incident where an alien AI that is a ship 'collects and accumulates' various entities it encounters and they have to coexist & travel the universe digitized in its memory banks. Forever.
Brin & Cherryh are two of my favorite sci fi authors, so forgive my shilling. But, I feel certain these will scratch your imagination itch regarding inter-species communication & interaction.