If you play in A Dorian and then switch to the G major (ionian) shape, you are still playing in A Dorian, or G Ionian - they're the same notes. You can play B Phrygian licks over a chord progression in the key of G major. You can try F# Locrian, but I can't recommend it... ;)
The secret to getting the sound is to emphasise different notes, so for example if you want to get a Lydian sound, ham up the sharp fourth... You should be aiming to get a modal sound rather than just using different shapes, and that comes from the theory behind the modes. Look at how the intervals of each mode relate to the parent major scale.
If you play the same lick in A dorian over a G major chord or a A minor chord, the result will be different. If you're improvising, some notes will sound more resolved than others... and that will change depending on the progression... experiment... :)
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