Help! Power question

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. 

+1 gs5556 (He would know)!

I agree with everyone saying get an electrician over to sort it out for you. But in the meantime you could try a line conditioner like this to allow you to work around the issue: 
https://tripplite.com/1800w-120v-power-conditioner-automatic-voltage-regulation-avr-ac-surge-protection-6-outlets~LC1800

(get the right wattage size for your needs there are multiple models)

I use one myself and while I don't have any issues I do hear the relays in the line conditioner click on and off every time my AC starts and the LC "boosts" the power.

thanks all again
re: the tripplite mentioned, many reviews says it has a rather loud "hum"