別に今日だって約束してたわけじゃないし
さっそく感が恥ずかしすぎる
In the second part, it seems like she's saying it like.. "the feeling of impatience is too embarrassing?" I'm too embarrassed of my impatience?"
Edit: From context she wants to see her crush as the talked before and said they'd see each other again after her work was done. Now she wants to go see him, but technically speaking they didn't promise to see each other a particular day.
さっそく is an adverb/no-adjective that basically means "without delay" or "promptly" when used positively:
But in a negative context, it can have a connotation of "hastily", "too soon" or "skipping the usual preliminaries":
This 感 is a suffix. Traditionally, it follows a fixed set of difficult kango (e.g., 罪悪感 "sense of guilt", 多幸感 "euphoria"), but informally, it can follow various words or clauses to say X-kinda feeling, X-ish vibe or X-ness.
When combined, さっそく感 forms an informal noun phrase that refers to a sense of hastiness or being rushed. さっそく感が恥ずかしすぎる means that someone's statements or actions are so hasty or eager that they become embarrassing.
