How big is the litter tray? Most of us use under bed storage boxes, they're big and deep so they can dig without getting everything everywhere. Straw is perfectly safe for rabbits, you can put it in their bed area and they'll happily snuggle into it. If you're using it for litter, though, make sure it's short-cut as it's more dense than normal straw and therefore more absorbent. The wee sinks to the bottom and the smell gets trapped down there with it, I find. I change mine every second day and it doesn't smell at all.
If it's just regular straw you have access to, you could use that too as long as you don't mind emptying it once or twice a day.
Other options: yes you could try kitty litter again as long as it's dust free, non clumping and all that good stuff. I have used the wooden pellets before and it was actually brilliant. I only stopped because it's heavy and we don't drive and with three rabbits and two guinea pigs, one bag didn't last very long. Or you could stick with your newspaper, but I would definitely advise to change that more than once a week. It is absorbent, but with only hay on top, there's nothing keeping the smell from escaping. Whatever you use as a litter, always keep the hay nearby and make sure the tray is big enough for them to sit/lie in it comfortably. They're cute but they're also gross, they like to 'go' where they eat. Just make it a nice place to be for them to be and they'll naturally want to spend a lot of time there.
As for the vinegar, I would increase the amount of it. I personally use pet-safe disinfectant spray on the litter boxes, I'm comfortable with that because I make sure to do it away from the animals, I keep the window open clear the air/get rid of any fumes and I rinse with lots and lots of water. They don't suffer for it. With regards to carpet stains I simply use soapy water. But it's up to you. How do you dry it? It may be that some of the smell is old water. If you can move that bit of carpet around I would hang it outside, let it sun-dry.
Oh and I can't tell if you mean you're refreshing hay once a week? You should be giving fresh hay every day. It would save you time and money if you just used normal litter (straw, newspaper, kitty litter etc) and put a handful of hay on top for eating, topping it up when it's low. That way it's always fresh and they'll eat it instead of weeing on it.
ETA: it is true what people are saying about intact rabbits being smellier than fixed rabbits, though. It's just a fact, I'm afraid, male or female.