FWIW, many folks around here would say your levels were fine before you bumped them up...especially if recording in 24-Bit. Get friendly with the volume envelope feature. Audition allows for non-destructive editing so you can play around with it and always go back and try again if you don't get it right the first few times.
Personally I never normalize my recordings. I generally will use the amplify gain and hard limit features when I am mastering my shows. Keep in mind you will introduce compression if you are trying to boost the gain over 0dB while utilizing the hard limit feature. I use it sparingly.
When I record I generally like to keep my levels around -10dB with peaks around -6dB. Sems to work well for me but everyone is different and I'm sure others will chime in.
Just my .02