It can be a huge problem and I suspect you can only diminish the frequency of the occurrences.

1) eliminate the smell on the items that have been peed on, on carpet you might be out of luck but on towels, linens, things you can throw in the washer... detergent and "percarbonbate" might remove the odor.  Remember that cats have a much keener sense of smell than we do and they will be attracted to pee on a surface they've done that before.

2) give a loud warning if you see "precursor" behavior, stopping to sniff an area especially if you see the "flehmen" (where the mouth is slightly open, they look like they are sneering and they are actually drawing scent into their nose and the VO organ at the top of the mouth

3) reduce opportunities.  Closing doors (which you indicate you try to do) is a good way to cut them off at the pass.  With the bathmat can you hang it up on the shower rod between uses?  If you can't close the bedroom door you could put a sheet of impermeable plastic over the bed (they probably won't want to pee on such a surface but if they do it won't get on the sheets)

4) reduce the stress... moving is very difficult for cats, probably giving them more attention (provided they aren't telling you guys to go away), more food and drink, a bunch of cardboard boxes, extra play time, extra treats.... anything to reduce their stress level

The kitchen mats probably aren't as interesting if they aren't as soft or absorbent.  If a cat feels the need to puke or pee out of the litter box they don't want the stuff splashing up at them, so they like a good soft surface that will take it right in.

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I've had issues with cats peeing out of place for several reasons:

1) territory squabbles / marking

2) UI infections

3) senility / old age
...

In general I'd suggest checking for a UI tract infection, but if this is a moving it is most likely stress from the moving.

Good luck