I usually do this when I am determining if it's an amp or a guitar issue. One step you have already kind of figured out.
1. With no guitar plugged in and the amp on, how much noise does it make at normal volume level. Other than hiss, it should not hum. The input jack should ground out with no input, hence why you should not have any hum. If it hums, attempt to clean the contacts on the input jacks.
2. With the amp on and the guitar plugged in, when you touch the strings, does hum diminish or go away? If it does, your guitar is wired correctly. If not, then you have a wiring issue with your guitar.
3. Is it a location thing? Does the issue occur at all locations, or just here or there? If so, it could be dirty power, light ballasts, light dimmers, or some other device connected to the same power source that is causing electrical noise to become amplified by the pickups or the amp. Power conditioners WILL NOT fix this UNLESS you buy a higher-end model and it ONLY addresses the power source issue, not the noise that is in the air from lights ballasts or microwaves. The basic Furman models do not have noise suppression in them, which are roughly $250 to start. The Merit series and other $100 models from any brand will not help.
Generally, if it is multiple amps that start having the issue together and suddenly, it is a power thing. It is usually due to something new added to the circuit in the house like a new light, air conditioner, or whatever it may be. Single coils are great antennas and if there is any noise in the air, they will pick it up. I have never had an issue with lack of shielding in my guitars, but perhaps I am just lucky? I don't like layering copper tape in my cavities and prefer the shielding paint. While shielding does help, it will not take you from noisy to silent, it will only take you from noisy, to less noisy.