You aren't wrong that 急いで is an expression in its own right, but 急いでる is here a contraction of a conjugated verb used in casual speech.
急ぐ is the relevant verb (meaning 'hurry'), so the て-form is 急いで; so, the ている conjugation is 急いでいる (lit. 'I am hurrying', more naturally 'I'm in a hurry'). But, because it a bit of a mouthful to pronounce all of those sounds casual speech, the い is often dropped, leaving you with 急いでる.
To answer the question directly, then, 「急いでるから...」 here really holds the meaning of "I'm in a hurry, so...", which fits with your interpretation.
It's sort of equivalent to saying "gonna" or "yeah" instead of "going to" or "yes" in English: not necessarily "proper" but totally understood and accepted/preferable in the right contexts.
See e.g. this article from Maggie Sensei for more examples and similar conventions: https://maggiesensei.com/2016/03/28/casual-contraction-dropping-い-i-てん-ten-とくtoku/
Hope that helps.