To answer your first question, existing melamine cabinets will typically have completed their off gassing process within three years of installation (plus or minus depending on a variety of factors such as how well ventilated your home is, were they uncrated and allowed to off gas prior to being installed, etc.).
- Recognize that it is not just the melamine that may offgas.
- Melamine is simply the exterior (and interior) finish of the cabinetry.
- There may also be particle board and adhesives to contend with.
You can hire a professional service company to test your cabinets, some mold inspection companies have a specialty in testing a variety of indoor air quality contaminants.
You will need to do some sleuthing in your geographical area for companies that perform this specific testing service.
Painting the cabinets
With respect to your second question -- what you can use resurface (paint) the cabinets --there are melamine paints and primers formulated especially for application over melamine.
- Both are high VOC products and must be applied in a well ventilated environment.
- The trade off is do you pitch potentially perfectly good cabinets into a landfill and incur the cost of new cabinets or do you expose yourself to some smelling, noxious primers/paints?
- This is a decision only you can make.
Given your geographic location, if you choose to prime/paint your cabinets you will be best waiting for late spring or early summer to tackle this job so that you can apply the finish to the door and drawer fronts in your garage with the door wide open.
Additionally, by allowing the doors and drawer fronts to 'cure' outside (but not in direct sunlight), you can accelerate the off gassing process from the new primer/paint.
You can also consider having the cabinets professionally refaced.
Consider the safety of the cabinet interiors
Lastly, are the interiors of your cabinets safe? Probably, but there is no way to guarantee because every person's sensitivity to chemicals is very, very different.
One safeguard to consider is to never place consumables directly on the melamine - always place food in a bowl, jar or some type of container.
Take care not to place acidic foods such as bottles of vinegar directly on the melamine if the bottles are prone to drips which will then sit on the melamine and potentially damage it after extended exposure.
For more information:
Read "How should I deal with off-gassing from a Melamine desk?" a Q&A answered by Joel Hirshberg.