Sure seems like bypassing at issue. Right at processor pins, Vdd to Vss, should
be a bulk cap, like a polymer tant, and a ceramic disk, .01 to .1 uF. Look at cap
datasheets, not all caps of same value exhibit good esr.
Take a look at LM317 response in datasheet. I looked at 78LXX series, in the
two datasheets I looked at no transient response info.
Regards, Dana.