Shout out to Amoz’s comment as it triggered some insights and funnily enough the answer hit me as I dozed-off!
With hell you are
LUST
You are
_ust
General formula is:
The clues in their first half hint at a number, which one should convert into a letter using the A1Z26 cipher and place it in front of _ust to make a word which is hinted at in the second half of the clue.
I'm not generally romantic, but
Show me a couple and I'm broken.
_ust isn’t generally romantic. ’Couple’ hints at 2 and letter 2=B and B+ust=BUST (I’m broken).
Two couples and I'm gone completely.
Two couples means 2+2=4 and letter 4=D so D+ust=DUST (‘I’m gone completely’).
You might think I'm very old:
Unlucky, like paper,
Unlucky hints at 13 and 13=M M+ust=MUST(in the sense of ‘Mustiness’ a quality often associated with paper (like paper) and with something that is old).
Or a little further on, like metal.
R is a little further on in terms of the alphabet position than M is and R+ust=RUST (like metal) since rust is Iron Oxide and it is also associated with something old.
All right, you've had enough!
Push with two hands, I'm on the edge,
2 hands = 10 fingers and letter 10=J J+ust=JUST(‘I’m on the edge’ or almost).
One more and you'll be rid of me.
3 hands= 15 fingers and letter 15=O and O+ust=OUST (you’ll be rid of me).
If you give me hell, what am I?
Thanks to Oehm’s suggestion in the comments:
‘Hell’ points to its soundalike: the letter L so L+ust=LUST!