You mentioned above a WP site and Pretty Link, so i assume (maybe falsely lol), that you promote your site and not direct Amazon urls. In that case how would that random string trick be able to affect the Amazon urls? I guess you worry about being credited for the sales.
Another thing: i don't promote Amazon, but i know that they're pretty anal about link cloaking/shortening:
"(v) You will not cloak, hide, spoof, or otherwise obscure the URL of your Site containing Special Links (including by use of Redirecting Links) or the user agent of the application in which Content is displayed or used such that we cannot reasonably determine the site or application from which a customer clicks through such Special Link to the Amazon Site.
(w) You will not use a link shortening service in a manner that makes it unclear that you are linking to an Amazon Site."
Whatever the fuck this last one (w) supposed to mean. Unless customizing, how are you supposed to make the destination of a shortened url clear in itself? If it's in context (and you're not purposely misleading), it's clear. If not, then not.
On the other hand Twitter can follow redirects, so generating unique shortened urls for the same long url might not do a thing, because they're only visually different, the destination stays the same. The random string trick changes the destination though.