Congratulations on the birth of your beautiful twin girls 
Caring for just one healthy baby is exhausting, and I can only imagine what you are going through, mothering your babies. Well done.
I think what you are describing is much more common than people let on, sadly. I think that the reality of becoming a dad for the first time is really shocking for some men, even more so if there has been a traumatic delivery. And while mothers (aware I am wildly generalising) just have to get right in there straight away, irrespective of how shit they feel, truly learning overnight the meaning of selfless service, some men flounder in the time after birth, for all sorts of reasons, and fail to 'show up' as the fathers and partners the new circumstances require them to be. They may or may not be aware of what is behind it, but selfishness, a reluctance to place the needs of others ahead of their own for a sustained period, a misplaced sense of entitlement, jealousy, an inability to work through exhaustion, insecurity, not knowing how to handle that they don't know how to do it, not wanting to ask for support, resentment of a change of spousal priorities, to name a few, seem to be common factors.
I see you have tried to approach your DP in a variety of ways with little success. I think you are unlikely to see a change until either
a) he has some kind of epiphany to relieve him of this character flaw, and which switches him on to a more intrinsic sense of motivation to do right by you and his children, or
b) things start to get a bit easier in a few months and it stops being too strenuous or difficult to participate more fully in parenting
I am not sure a parenting class will bring about the change you hope for and jolly well deserve, it runs deeper than that; he needs to appreciate the complete inequality of the set-up he is forcing on you, understand that the normal limitations of physical capability also apply to you, and that going to work everyday is a walk in the park compared to caring for just one baby around the clock with minimal assistance, let alone multiples or a poorly baby. Mothers with partners like these are forced into an impossible situation where they have to carry on caring as you can't just go on strike, and so are left carrying a huge emotional, physical, logistical and psychological load. And it is so much more difficult to do when one's spouse is unkind, critical or dismissive to boot.
I have known so many male partners, apparently competent, intelligent and committed, become first class pricks when their babies are born, even if they like to keep up appearances to friends and family, and their long suffering wives and girlfriends don't want to blow their cover 
My own DP took a while to get his head around the stark realities of parenthood, and although he got there in the end, I can never unsee the deliberately useless incompetence of his early days months of fatherhood, and I have definitely lost some respect for him as a result.
I wish you all the best, and lots of courage and strength.