It sounds to me like you have an issue about roles on your hands, not one about clothing. Here's my analysis:

I've said to him before that if I fold his clothes up, will he put them away? He said that he would. He hasn't done it - once ever!
You've baragined with DH for him to do part of the washing job - do you feel like its your job and you are negotiating with him to help you? Or do you think its a fair division of a job that "just needs doing"? Did he get any say in whether he would put them away? Do you think he thinks you should just do the whole job?

'They're mostly your clothes you know - Jumpers and pants' I remember I washed 3 jumpers and 3 pairs of pants (in preparation for winter) so it's not like it's a whole lot - but is it too much for him to just put them on the chest of drawers so I can put them away later?
Does he feel like if he does "his jobs" and you should do "yours"? Does he resent having to do "his jobs" (because you are home full time? because you are the wife? because he does other things?)

Now, the fact that I've already asked him before if he'd put them away is why I think I have the rights to be annoyed at him. But it's every time I do the washing! It's always 'Why haven't mine been put away?' and 'When I say to put the clothes away, I mean all of them'
Have you assumed the role of mother here? (in that you chose the jobs and the kids are supposed to do them). Has he taken any real responsibility for his part in running what is HIS house as well? Does he feel he should? Is he the child who needs looking after, or the "man who works" who "should" be looked after? Does he care if things get put away? Do you have a different standard of cleanliness that him?

Whenever DH is on holidays from work (he's a teacher) he does housework, but each time he does, he makes me feel like garbage - like I don't do anything while I'm home. He gets home after work and says things like 'Ugh, this place is a pig sty - I guess I'll just have to clean it up on the weekend'
Does he think you aren't pulling your weight? That you are home full time and should do more? OR have you led him to a position where you think you should do it all then get cranky with yourself that it isn't done? Has he ever looked after a baby full time? Does he realise what is involved in doing this?

I know this is asking more questions than providing answers, but it seems that you tow have a disconnenct in expectations - both of what is possible to get done in a day with a 5 month old AND what each of your new "roles" actually entails.

There is no easy or correct answer to this sort of thing. The minute we left the eara where the wife stayed at home and did EVERYTHING and the man worked all day and had wifey bring him his pipe and slippers in the evening, we threw out the "rule book" - it is literally for each couple to work out between themselves. The only thing I know is that you will never get to the "answer" without first asking these sorts of questions......