It's not going wrong at all, your clever baby is doing exactly what he needs to at this point, namely tell your boobs 'yes! Yes! Milk milk milk!' and also to stay close to his dear mama who is all he's known for 9 months. Congratulations on your new arrival.
Being at the breast is about so so much more than feeding and being satisfied or full up. He's priming your supply of milk, stimulating a release of oxytocin (the 'love' hormone) to bond with you, and staying close to your smell, your heartbeat and your voice. Try to think of it as nursing rather than feeding as there are associations with timing, schedules and being 'full up' when it's considered a 'feed' alone. Being fussy also ensures lots of attention from you - clever, clever baby!
Is he producing wet & dirty nappies? Has his poo changed colour from meconium to brownish then yellowish? If yes, those are good things. If you're not in any pain or discomfort this is also good. This is a time for you to sit/lie down with your phone/remote/laptop/book/water/chocolate and baby and just get your milk on! You're doing so well.
If during a session you hear/see deep rhythmical sucks/swallows (sometimes difficult, I know), and he takes himself off the breast, he's almost certainly satisfied. If he falls asleep that's also good, though be aware if he's falling asleep frequently he may not be transferring milk as well as he could.
Lots of babies are more fussy in the evening, and may 'cluster feed' (basically exactly what you describe - constant need to be at the breast, but not seeming that satisfied when they're there!) - again this is instinctive and evolutionary. Night time is dark and scary. There might be bears and wolves, best make a nuisance of myself so mama holds me close, thinks your baby!
Have faith, stay in touch with Mumsnet (this forum is fantastic), and if in doubt you can always call one of the BF helplines. I'm so sorry i can't link to them from my silly phone, but NCT, La Leche League etc have a line, and there's a national service too.
Congrats again. Keep up the good work!