I have the same issue I think.
I am using Supply Theme. Support and I changed the code to always show quantity left not just when it falls below 10. When I choose a variant, Price is properly updated immediately but quantity shows the amount of the last variant unless the screen is manually refreshed.
The new-variant quantity update should be done at the same time as the new-variant price update.
Support fixed this issue for me. I now always see quantity in stock on my Product page and it shows the correct variant quantity without having to do a refresh.
Because Supply is a free Shopify theme, they did the coding on the refresh bug for me.
The coding to have quantity always show instead of when stock is less than 10 was very very simple and just involves putting a comment and endcomment around a couple of lines of code. While they did it for me I could have done that one myself with a couple of simple instructions.