I think IKEA has some cabinets that are mostly metal. I wonder if that would be good? Alternatively, locally made custom cabinets might be made of healthier materials (real wood) and sometimes apparently are quite reasonably priced.

You could do a combination of open-shelving and cabinet design using your old cabinets. My inlaws have open shelving for upper and lower cabinets. You could definitely build them on a budget, and possibly use real wood coated w/ a natural oil finish. Depending on the look you are going for, you could do open shelving for the lowers as well, using wood or sheets of metal.

Have you looked into local custom cabinet makers? I had a local guy build all mine from ash, it was 5200 for a 12x12 kitchen plus an 8ft long wet bar.

I finished them myself with a water based stain and a water based poly that wouldn't stink up the house and give me migraines.

The only smell I had was fresh cut wood.

Have you considered Barker Cabinets? I've included a link to their specifications. Barker builds Ready-to-Assemble cabinets made from all USA components. There is no particle board, they use Purebond formaldehyde-free plywood so there is no formaldehyde off gassing. They also use very good hardware for the drawer glides. Their cabinets are available ready-to-finished, or finished with low VOC finishes.

Several people on this forum have used Barker to put together beautiful kitchens, at great savings.

Here is a link that might be useful: Barker Specs

athomesewing beat me to my suggestion. We are just finishing up our kitchen with Barker Cabinets. I can tell you that there is absolutely zero "new cabinet smell". My MIL is extremely sensitive to odors and very easily gets headaches when she is in a home with new surfaces that off-gas...carpeting, furniture, curtains, etc...she has had absolutely no reaction when she's been at the house recently. On top of that, Barker's cabinets are absolutely top notch quality and absolutely beautiful. Yes there are cheaper options out there but I could not find a cheaper option for the quality I wanted than Barker.

You mention cost as a factor. Completely relocating a kitchen is going to be expensive. Even moving the sink or range would mean moving the plumbing, electrical and/or gas lines.

"Standard particleboard core casework and tops, made with particleboard meeting ANSI A208.1 and properly laminated and edgebanded, should easily surpass formaldehyde emission requirements."

Here is a link that might be useful: Emissions suppression