Sounds very difficult but the good thing is that both parents are at least talking about it. I agree 50:50 provides an escape route for him - e.g. from chores he doesn't want to do, but if both parents also agreed to the same sanctions (so to speak) then life at House A would be no more advantageous than life at House B and that might just help him to grow up and start accepting responsibility.
Yes - we all know it's particularly hard to find work ATM but to not even try is taking the piss - especially if he thinks board is "unfair". Refusing to look for work at all can't be allowed to proceed long term - does he have any long term plans ?? ... or does he literally believe he can doss about at either house and be fed with no contribution from him at all ? Assuming however, that right now, neither parent wants to literally see him on the street (given his age, given college didn't finish that long ago) then until he gets more work damn right he should be contributing at each household in other ways by making himself useful and doing stuff which would alleviate some of the stress for the people who are supporting him, and which would show that he actually appreciates he's an "adult" now who has no automatic right to unconditional support.
Well, that's the theory but fully appreciate how hard it is to literally "make" him do chores. If appealing to his conscience doesn't work then as I suggested earlier I think the next step has to be identical sanctions which are stuck to rigidly - a.k.a. making life as unappealing for him as possible. That means (or could mean, you can probably think of others) stuff like: basic meals (he gets no "treat" stuff like cake, biscuits, icecream, sweets, crisps etc - might have to hide this), no laundry done for him, no lifts anywhere (why should he get favours?), no internet access (password family PC, password modem etc), gadgets that don't belong to him personally removed from his room (e.g. TV, playstation if applicable) ..... some people might even suggest removing his door - and therefore his privacy - until he "earns back" the right to be treated like an adult, e.g. when he starts acting like one.
And long term, however hard it is, I think both your DP and ex need to agree an approach if he continues to be so entitled. That might mean a threat of asking him to leave if he doesn't pull his socks up before a deadline. I know that's very hard to think of but it's completely unfair he expects to be carried while making no contribution of any sort to either household. Would be entirely different if he had plans, was making an effort, was doing voluntary work (so arguably getting useful experience and/or contributing to society), was doing housework/garden.