Oh lovey.
First of all a massive CONGRATULATIONS.
Secondly, the early days of twins are HARD HARD HARD HARD. I don't think anyone will tell you otherwise, and if they do they're lying. It is survival. Do whatever you need to get through, and don't feel guilty for one tiny iota.
Tandem feeding is REALLY HARD. I only managed it successfully for a small window of time, basically once they had a bit more strength in their necks, but before they started wriggling. There were plenty of evenings when I would get them on and then be completely beached on the bed waiting for DH to get home from work and help me get them off/winded etc! Have you got a proper twin feeding pillow - they're not the answer to everything, but they do make a few elements of it easier - check out here to see what I'm talking about.
Other good sources of information are the Mothering Multiples Book and La Leche League (0845 120 2918) - I had a telephone consult with a multiple bf counsellor as there wasn't anyone local to me, but I live in quite an isolated place so you might have one who can pop round.
Were they term? Did you leave hospital with the midwives/your hv happy with their latches? If you can successfully feed each singly I would really focus on that for a while, one can always fall asleep on the boob while you sort the other one out - assuming they don't wake absolutely in sync. The other option of course is bottles at night, but I would really really try and persevere with ebf if you can, you will so reap the rewards a few weeks down the line (besides all the benefits to your DTs of course!)
Is your dh still on maternity leave? If he can extend it with any annual leave I would strongly recommend it, these early days you need as much support as you can get.
As for the falling asleep with them on your chest.. yep, vividly remember that. I also remember shouting at DH as he couldn't stay awake holding them at all, and he is a heavy sleeper, that scared me. I think rather than fight it it's better to work out ways of doing it safely. Do you have a bedside cot you can roll them into easily? Get a bed guard? Move DH into a spare bed if needs be? There's lots on here about safe co-sleeping - maybe wear thick pjs and put your duvet right down below your waist so they can't get under it etc.
Sorry that is an epic post and you probably don't have time to read it all, but just to say you're doing AMAZINGLY to be tandem feeding, it will get easier. Don't be daft worrying about 'stimulating' them at this stage with songs etc - they just want to eat, sleep and be cuddled - you're doing all that, you're a brilliant mum.