I do take issue with newsletters that don't give an opt-out option, by principle.
That said, Aweber is a legit company; this can be confirmed by doing a simple 'net search on them. I've been on a few mailing lists managed by Aweber, and all I've ever received was email that I'd explicitly requested and/or approved. I highly doubt they've ever shared any of my information with anyone. I've signed up for newsletters and the likes with a made-for-spam email addy, and my spam has never increased afterward.