Really don't know what "copper keeps coming off the board" is supposed to mean. Presuming standard copper laminated FR4, the copper foil is rather tightly bonded to the substrate. Having unsuitable etch resist, the copper might full of small holes, but still won't "come off the board". On the other side, even small copper particles of an incomplete etched board stay steadily.
Copper traces will however easily delaminate when soldering it too hot.
But as said, I'm talking about standard laminate quality.
Also presume proper rinsing of etched boards to remove any etchant residuals, otherwise copper might be attacked later on.
Coating copper traces might be useful to avoid brown or black tarnish, but not necessarily required.