Send me a bottle of odor in question and have the contractor send me a sample of mastic he used. I'll sniff and compare then let you know if you are smelling mastic or something else.

mastic as far as I can tell doesn't have a smell and is typically hard within a day or so. Most kinds are similar to that of caulk.

In some cases in building returns and such I've used spray foam to seal the new return to the sheetrock for a good sturdy air tight seal. While you could use mastic for this purpose it's much harder (if not impossible) to get mastic into tight spaces to seal properly.

Spray foam will give off odd smells for up to a couple of weeks depending on how much leaks thru to the duct side that feeds the home, until it fully cures.

What you can do to alleviate this is turn your fan to your AC to the on position. This will prevent the smells from building up too heavily within your duct system until the curing process is over and you may speed this process up.

Try that for a couple of weeks to see if that solves the problem. After which you can then set the fan switch back to auto if you prefer.

Our hvac guy had told us in the past to never run the fan in the on position because it can cause mold in the plenum. Are there any other things we try to hurty along the curing process.

I would think the opposite would happen. Running the fan would dry out the coil faster and reduce the chance of mold.