Does Romanian distinguish between blame and a blameless apology like some other Romance languages do? e.g. in Spanish:
"Lo siento que se le tiró el café". - I'm sorry that your coffee spilled (the person speaking had nothing to do with the coffee spilling).
"Disculpe por tirarle el café." - sorry for spilling your coffee (the person speaking was responsible for the coffee spilling).
I'm going to give this a shot. I could be wrong, because I grew up speaking Romanian but have never been to Romania, so I speak very colloquially.
Scuze is not a verb, so you can't say, "Eu scuze." Simply "scuze" is fine for "I'm sorry." Another thing you can say is "scuza-ma," which I think is more of a request ("forgive me" or "excuse me," rather than "I'm sorry").
If you want to be really apologetic, I don't think you can expand on scuze, but you can say "Îmi pare foarte rau" (there's the little accent on the a in "rau," my keyboard is dumb), which means something like I feel very (foarte) bad
My experience with Romanian speakers both in the UK and Romania would confirm that 'scuze' on its own can mean: Excuse me (I'd like to go by you or briefly obstruct your view like at the cinema); Excuse me or Pardon (I didn't hear or understand what you said), Sorry (both as a mild apology and as a polite apology for bumping into someone), and as Excuse me (I'd like to attract your attention or speak to you).
Someone knocked my coffee off the table once and although I'm not sure exactly what they said it was way more than 'Scuze'!