Not knowing what compressor you have, I will say all are not equal, especially when it comes to accuracy of the gauges. I have a Dewalt compressor that has good quality gauges on it. I set mine to 40 psi for blowing the RV lines. The other gauge will show 165 psi. It uses a big volume of air when opening faucets, and my line pressure will drop to low 30s while blowing, go back to 40 psi when faucet closes. Your compressor line set psi should not continue to rise well past set psi.
I doubt 60 psi for a short while, would cause any issue with your lines/faucets, but again, all lines/fittings are not equal. Here is the compressor I use. The hose fittings, gauges, switch, and valves, are good quality.
Jerry