Basque Surname Etymology Help

(disclaimer: the words in surnames might be older words that are no longer commonly used in the Basque language, so if there's any discrepancy with the translations, that might be it; but please tell me if it's incorrect regardless)

Hey y'all

My surname is Azaldegui (from my grandfather who I unfortunately can't ask and don't have any current ties to any family in Basque county), which I've scoured the internet for any etymological meaning; The most detailed I got was getting on an official Euskera surname storage database site, which listed it as a combining of Azeri (fox) + Tegi (denoting place, usually meaning where the person/family lives/lived near to or at), or place where the foxes are/near the fox burrow

although this hasn't exactly sat right with me, as when i dug deeper on grammer rules for Basque/Euskera, if there's an l or n ending the word, '-tegi' becomes '-degi', and azeri doesn't end with either of those letters.

However, I recently found that cabbage is Aza in Basque, and so my surname could potentially mean place of cabbages, or near the cabbage patch (irl cabbage patch kid ??? 😳); I'm not exactly sure how the l ending would get there but I'm by no means a grammatical expert in Basque. -- edit: extremely less likely now, especially because there's almost no way an l would be randomly put in there just for a surname, and there's an actual word for azal (update for that posted below)

I also found a Basque town called Azaldegi, but that didn't have an etymological meaning listed from what I could find.

(Not sure if it's exactly accurate, but using google translate, 'azal' translates to 'skin', which potentially makes my surname mean 'place of skin' which: terrifies me ngl) -- update: 'azal' does translate to skin, but more broadly it means the 'surface' of things, such as the bark of a tree or surface of a calm sea, those are the 'skin' of those objects; thank you u/ZakjuDraudzene for showing me Elhuyar Hiztegi !

Does anyone know if I got the meaning correctly in one of these guesses, got close but no cigar, or went way off the deep end? Any help would be greatly appreciated, thank y'all regardless!