I played through the game solo on legendary and in the game it shows that I have beaten all of the levels solo on legendary, but the game is not giving me my achievements.
Hey Dragon!
Does the achievement display “Done! Unlocking” in the Xbox One achievements app? If so, you simply need to wait, and it will unlock eventually.
Another issue could arise from improper stats uploading. According to your MCC Service Record, you are missing Legendary Solo completion for Forerunner, Reclaimer, Shutdown, and Composer. You might want to recomplete these missions, playing each in one go and online the whole time. Sadly, I can’t recommend much besides that. Hope this much helps, at least. 
Same problem here. My in game says all solo legendary, nut this site says co-op for 3 of the levels…