I don't think you have to pot it on the turn. The best draw against you is a set with an OESFD, so 11 outs. That's not very much with only one card left to come. You can price even that hand out with a less than pot sized bet. Without potting it you will probably lose some value against lower flushes, so it's a tradeoff. I'm not certain which is exactly right, and the answer would vary depending on the looseness of your opponent (tight = less than pot, loose = pot).
You do have to raise the turn, though. I fail to see any hand the opponent could have for which flat calling is better. You give draws a chance to suck out for free. You let lower flushes or straights hit scare cards to kill your action. It's only good for a total
bluff, which is pretty damn rare in Omaha.
I almost always pot the river if I'm betting. There are no more cards to come, so the odds don't matter (ok, this is not true if your opponent is putting you on a range of
hands with different probabilities, but that's too advanced for 90% of players). If he's calling, he's likely calling any bet. It's a principle that a lot of players exploit in hold 'em, too.