Eight ohms on 5V = 5/8 amp = 625mA. If your wall wart can't deliver at least that, you'll have trouble. You need more current to supply the regulator. It can't output more than what's it getting in. You can also try a bulk electrolytic on the input side to reduce ripple, but a more stout wall-wart is the first order of business.
If you have good DMM, just look at the AC voltage on the 5V rail without the speaker connected. It should be negligible, although you have to have a good DMM for it to have a wide frequency response band.
I have only ever seen oscillation on a 7800 series regulator once, when used without output capacitors, and it had an oscillation frequency over 1MHz. A cheap electrolytic on the output fixed that. I seriously doubt you have any oscillation problems, and all of your problems are caused by hooking the speaker to the 5V rail.