SummitAppliance.com has some shorter refrigerators listed along with less depth.

Otherwise, it shouldn't be THAT big a deal to revamp the current cabinet over your old fridge's space, I would think.

Oops, I forgot the obvious...get the old refrigerator fixed.

Try the Fisher Paykel fridge, E522B. Here are the specs:

Height - incl. feet and hinge cap: 66 3/4
Width: 31 1/8
Depth - including handle: 28 7/8
Depth door open 90°: 56 7/16
Depth - door closed not including handle: 27 3/8
Minimum air clearance - each side: 1
Minimum air clearance - at rear (incl. compressor tray):1 1/2
Minimum air clearance - on top: 2
Refrigerator Storage Volume: 12.4 cu.ft
Freezer Storage Volume: 5.2 cu.ft
Total Storage Volume: 17.6 cu.ft

So you'd have to cheat a bit on the top air clearance. It's a nicely organized fridge, we almost bought it but decided it was a wee bit too small.

Here is a link that might be useful: Fisher Paykel fridges

had a similar problem once and a skilled carpenter was able to modify the cabinet b/c it had a facie board above and below. There are more options now on counter depths but then the brand and feature choices rachet down.
really depends on the cabs of course, but i agree that it might be just as easy to pull that ONE cab thats over your R/F and even go to open shelves (country style). Or thats the home for your new ceiling suspended pot rack.