normally, *across* the 2 relay pads (if you put a DMM across the 2 relay wires) you should see very close to 0. 5v thru a 47ohm and into one wire, out the other wire, into a 47ohm and back up to 5v. that should add up to zero
that assumes the ULN chips are sitting at very high Z until being told to 'short to gnd'.
maybe try other ULN chips? I use toshiba chips, I think. maybe the ULN you have has problems.
when you say you have hum now, that almost sound like the 5v supply is still straining. there is a connection between the attenuator ground and the lcduino/d1 logic ground - and it goes thru a ferrite wire/bead that I think I saw on your board. if you get hum in the audio path, your ground must be 'bad' OR maybe some of those probes you have soldered in place are picking up stray hum. I wouldn't worry too much about hum in the audio path when you have things set up in a test config with probes on it but I would worry if your ground is hummy.
a fancy supply is just not needed. the a10 has a small s25 7805 regulator that is a foot or less away from the lcd1 and the d1/d2 stack. if the regulator is putting out 5v and is kept pretty close and the ground is strong and also local (and not going all over the place) - and if the supply can handle 500ma or an amp as a surge at power up and maintain a few hundred ma during operation, things SHOULD work. people are not generally reporting problems if they keep to the design and not have overly long wires.
I've done lots of testing with just wall warts; but I always try to have the REGULATOR loop kept to a foot or less of wire. have whatever you want leading UP TO the regulator; but after that, I keep my ground short and my b+ also very short. when I do that, I can drive even 3 d1 at the same time (you saw the photo I posted).
so, try a different type of ULN chip (toshiba ones) and maybe that will help. before even doing that, does swapping some chips from other boards change things at all?