I found this. Its worth a try, although, your cabinets appear to be badly neglected.

Melamine Surfaces

1. ✓ Wipe with warm, soapy water on a soft damp cloth.
2. ✓ Follow all cleaning with a streak free glass cleaner on a soft cloth. ...
3. ✓ Apply cleaning products to the cloth rather than directly onto the surface being cleaned.

Thanks so much. Hopefully, it will help a little until we can replace them.