Do you have extensions in the supply line that go above the sink? If not you should since that appears to be the highest point in your system. It sounds similar to a water hammer problem that could be altered by the coiled lines under the sink.
I have had a similar problem and it was do to sags developing in the line out to the sewer. You get sludge buildup that effectively blocks the line. What I did was have the line scoped and ended up running a new line. A very ugly solution but, one that worked!