Help! Power question

Call an electrician. There is probably a loose lug connection in the meter pan or main panel on one of the phases.

Did a licensed electrician do the connection on your new ac unit? If not, you need to have everything checked. Refrigeration people do not have to be licensed electricians in my experience. Who knows what was diddled on the install. 

As djones51 mentioned, having a "hard start device" put on the A/C unit should be a first step in my opinion.  I think of it more as an "easy start" because it reduces the surge needed to start the A/C unit.

You could also put a "Heat Recovery Unit" on, which costs about $500 - $750 and will provide heat for water in the water heater.  I had one here in Florida and could turn off the water heater from May through October.  It might be considered unrelated, but the less the demand on the panel, the better whatever surge can be handled.  Also, this increases the efficiency of the A/C unit by about 0.5 SEER.

Finally, and this might not be approved of by the audiophile group, you could add an APC Smart-UPS 750 or similar sine-wave unit.  I don't really understand the technical arguments against this, as I think as long as the UPS can provide the power it should be fine.  Now, a little APC 350 might not be the best. I do think it should be a "pure sine" UPS, not a "stepped" approximation.  (I was able to run an "energy star" refrigerator on an APC-SmartUPS 750, but the non-"energy star" refrigerator wouldn't work because it created too much surge on startup, so the UPS shut down.)

The "surge protector" is just for "protecting" and not for "preventing."

Perhaps a Sola constant voltage transformer would solve this issue. We used to use these for isolating test equipment. I have seen them on ebay for not much. Like under $200 for a good sized unit.