They look fine, but after some thought about the meaning of what you are saying I personally think they do need the conjunction.
It is fine to use a comma between related statements, for example:
He felt hot with rage, furious that he'd lost the job.
The second statement is really a continuation of the first thought. The first statement is that he is angry, and the second statement is why he is angry.
Your example is about two different emotions - feeling sorry for oneself, and anger. They might be two emotions over the same thing - the loss of a job - but I feel they are two independent clauses because one feeling doesn't really depend on the other. You could feel one, or the other, or both.
He was feeling sorry for himself and angry that he'd lost the job.
It really becomes apparent if you take away the reason because you'd have to say:
He was feeling angry and sorry for himself.
It wouldn't make sense to say "he was feeling sorry for himself, angry". Likewise, it wouldn't make sense to say "He was feeling furious and angry" because they both mean the same thing!