Within the Gyazo settings, by right clicking it in your hidden icons section, you can edit the button within there.
instead of pressing the button on your taskbar or whatever, hold down CTRL+SHIFT+C for gifs do CTRL+SHIFT+G
Instead of clicking "esc" when trying to take the screenshot just click "t" so it opens the chat a bit bigger and it'll be same as paused like that and you should be able to open gyazo and take a picture. Due to I'm not sure why yours isn't working when using the hotkeys.
^^If you're taking screenshots of chat just press t before you gyazo When I want a screenshot of the game I take a screenshot with f2, open it in-game and screenshot that with gyazo