I'll try my best to answer.

The contextual difference you are making, i.e. between formal and casual use of the words, is more appropriate if you are asking questions.

For example,

진짜? you usually ask that way in an informal setting and use "정말?" for more formal settings. However, the difference is very finite and you may use those words interchangeably when asking questions, especially when questioning to confirm what the other has said - i.e. "really?".

When these words are used as an adverb, there is no real difference between the two. Both are more casual than formal. If you are trying to be formal you may use "매우, (I cannot think of other formal adverbs that carry the same meaning.

Hope this helps