When I first bought Skyrim: Special Edition for my Xbox One and started up a new game, I chose the Argonian race. It was nice at first, but certain gameplay and visual elements of the Argonian race started to annoy me after a while, like the big, ugly tail, facial features, etc.
Is there a way to change my characters race, in game, without starting over?
Try using the Cheat Room mod found here https://mods.bethesda.net/#en/workshop/skyrim/mod-detail/3010376
It can allow you to access the ShowRaceMenu command which should let you change your race.
Sadly, there is no way to change your character's race mid game, without starting over.
There are methods of changing your race as well as various other features of your character, but they're only available for the PC version of Skyrim.
I assume you play in 3rd person mode, as you find the tail to be quite annoying. Maybe you could read up about the Argonian race on the internet, to develop you interest in the race as a whole (I personally enjoy playing as an Argonian.) I recommend reading about the lore of the Shadowscales.
Base link for lore on Shadowscales: http://www.uesp.net/wiki/Lore:Shadowscales
I know that this is very very very late but for newer people looking you can just enter you console and type the command showracemenu you don't need any mods but you do need to be on PC
