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I'd always go with Brooks, Asics give me blisters too, but you need to go with what suits your foot best!
They are as 'good' as each other.
The Adrenaline is a bit more stable in the rearfoot but but the diffrence prpbably would make a diffrence to most.
So assuming they are both the right stability for - the 'best' one is the one that has the best 'fit'.
They don't have exactly the same fit and one will match your foot better than the other.
One will feel more 'glove like', contour your foot better, flex in the right places and generally feel less that you have something on your foot - less noticeable and possibly feels lighter (whether or not it actually is).
That's the one to go for.