Hi,
As Ron H pointed out you really should add those input capacitors, last year I was working in the tech department of a medical device company here in Brazil, and we were having similar problems with a power supply that used 7805, and adding those small tantalum caps, (before and after) the regulators really helped with the ripple figures, here is a quote from the application notes on the datasheet from national:
“An input bypass capacitor is recommended. A 0.1µF disc or
1µF solid tantalum on the input is suitable input bypassing
for almost all applications. The device is more sensitive to
the absence of input bypassing when adjustment or output
capacitors are used but the above values will eliminate the
possibility of problems.”
What type of capacitor did you use for the adjustment pin bypass?
Can you measure the ripple frequency at the output?
This is my first post here, hope i could help, and sorry for any grammar mistakes, English is not my native language.