The quote is grammatically incorrect, but not for the reason you believe.
Let's start with the first comma in the quote. When you have two adjectives in a row, usually the adjectives are supposed to be separated with commas.
So instead of amazing full sentences, it should be amazing, full sentences. This rule is broken very often, and it is OK to break the rule in most forms of writing today.
I believe the way you put commas around with proper grammar is correct, but both of the commas are optional. The sentence is grammatical with or without the two commas, and you should either use both or use neither.
Finally, in formal, written English, the sentences are ungrammatical, because really the two sentences should be one. The second sentence is technically incomplete because it lacks a subject. The writer did this intentionally to imply that he or she is watching the President speak (in the first sentence) and is pausing before making the observations (in the second sentence). It's a technique used to make the writing more expressive, but technically it's not correct in formal English.
So in formal, written English, the following are correct:
President Obama is speaking in those amazing, full sentences, with proper grammar, that we all love so much.
President Obama is speaking in those amazing, full sentences with proper grammar that we all love so much.
Normally, tweets aren't expected to follow formal grammar rules.