"I'm going to go to the store" vs "I'm going to the store" to communicate that I am leaving right now

Solution 1:

"I'm going to go to the store" = I will go to the store in the future; I am not leaving now. It would normally be followed by a time, e.g. "tomorrow".

"I'm going to the store" could be something I am currently doing, about to do, or will do in the future. It's not idiomatic at all for "currently", and it would normally be followed by a time: "now" or "tomorrow" for example.

Idiomatically, if I wanted to tell someone I was currently going to the store, I wouldn't use the verb "to go". So I might say "I'm en route to the store".

Idiomatically, if I wanted to tell someone I was about to leave for the store, I'd say "I'm off to the store."

All this is the highly subjective opinion of a native British English speaker (me). And, to be honest, if I'm talking idiomatic British English I wouldn't use the word "store", I'd talk about a "shop".