Hi Rosemary
9s are too light really unless you want to be able to bend real easy but you will have to sacrifice pitch stability IMHO. It MIGHT be why you feel that there are intonation problems as they deflect rather easily with the resultant change in pitch.
Fender strings should be fine. I know my Oppo in Perth (JJ) uses Fender all nickel strings and recommends them. I know that the early Shads sound is a composite of many things, pickups, strings, echo, amp, studio effects AND playing technique to name but a few but D-Addario and Elixir Nanoweb DO deliver.
When you are up at the guitar shop buy a light pleck - 0.7mm and that will help you to play chords much more effectively. "Strumming" is in the wrist and a light guage pleck will brush over and back on the strings and DEFLECT with wrist movements whereas my hard-as-nails 1.3 plecks just dig in.
cheers
Ian