Could I write a sentence like: "Adjective, rest of sentence"?

Yes, as long as the adjective is modifying the SUBJECT of the sentence. (In your sentence "I" am "ecstatic," so it works fine.)

You can't, however, say something like this: Dripping wet, I handed him a towel. ("He" is the one who is wet, not "I", so this doesn't work.)

yes.

i agree with Bedlam, i believe this structure is called participle phrases.