Low voltage under load from main line
It sounds like you have a bad connection or a broken wire somewhere upstream of your main breaker. Assuming that the power company's test were valid, then the problem would be the wire between your main breaker and the power meter. Other possibilities are that your main breaker is bad, or that the meter, meter box, or something upstream of that is broken.

See my answer to Insufficient voltage to dryer receptacle; what next? for why broken wires or breakers can cause weird voltage issues.