I am guessing if your DH is studying and revising you are shattered from childcare and broken nights. If you were both hands-on and sharing the workload it would still be a huge effort so no, you are not doing a terrible job.
You have just brought a new life into the world, and physically recovering. it could be PND, did you have it after your DS? Or it could simply be holding the fort on your own. Please don't hesitate to raise this with your GP, it is not uncommon and there is nothing to feel ashamed or embarrassed about.
Even with exams, with or without regular work hours on top, your DH needs a mental and physical break, that's understandable, but so do you. I hope he can still pitch in? He doesn't have to contribute a lot to make your life easier but he should still take some tasks on, pick up shopping, bathe DS or DD, or whatever. So often the person who isn't doing laundry or cooking something basic says, "Well, it only takes a minute, what's the fuss about?" whereas it can help give you a break even of half an hour at a time to shower, put your feet up, walk round the block for fresh air alone. Do you each get a lie in either Saturday or Sunday? Is there any one who could babysit for a couple of hours when DD is old enough to be left?
When do the exams finish? Can you hold that end date in mind and keep going? Without exams does DH generally pull his weight?
It wouldn't be surprising and no way any lack of mothering skill on your part if DS has been playing up more than usual. He would have noticed your pregnancy in the later stages and gradual limitations on close hugs and energy and of course, since his little sister was born, he's got competition for your attention. This is not unique to you, please don't feel bad.