Now I feel even worse If he just needs a little extra TLC at the moment. It's just so constant it's really getting to me.

I've tried putting him off by saying we can have milk after breakfast etc, but he just screams and cries until he gets the milk. Maybe I'm too soft?

I feel like such a sook.
First of all you are not a sook, such frequent feeding and interupting what you need to get done is frustrating and would try even the most patient of mothers.

My DS was also a frequent feeder, by 2 it was starting to slow down but on the days where he did feed as frequently as your DS was always the leadup to some sort of sickness, be it a cold, teething etc... By the time the sickness appeared I would be well and truly over it too and then have that uh-ha moment and feel that guilt too so I wouldn't be surprised if it was either cause he was sick or might be coming down with something or teething or all 3 at once

My other suggestion would be can you have some time out? even just a couple of hours without DS this weekend? just to give yourself some time (I found that if I wasn't around that he seemed distracted by whatever was happening around him that he was ok without me)