Part of the problem was that for a long time when linguists examined sign language, the users would automatically code shift to try to facilitate communication—so the linguists were seeing something more like signed English (or a pidgin—simplified grammar, for instance, to the point of having few features of a real language). Still, ASL has been recognized as a real and distinct language for at least 20 years by now.
I don’t know if ASL is considered a strictly a post-positional language. Generally modifiers follow nouns. But, the thing you were talking about, Maegan, is called “fronting” where, regardless of case, you can put the noun phrase that is the “theme” of the sentence in front. So a direct object, for instance, can be the first sign. (IIRC, in linguistics, they call this the “Theme-Rheme” approach. And I admit, it’s been ages since I learned this stuff, so don’t rely on what I say.) The nominative case noun phrase isn’t necessarily the “theme” or non-grammatical subject of the sentence—so you can put whatever is the theme in front. It’s done in informal English too. “Money—that’s what I want” instead of simply, “I want money.” (In the second one here, “I” is the grammatical subject, but the “theme” of both these sentences would be considered “money”.)
Oh—another cool phenomenon that Maegan’s post reminded me of: home signs. (Again, this is more about the way things used to be.) Unlike most speakers of a language, the majority of people who use ASL don’t learn it at home (because their families are very often hearing).
In urban areas, that works fine—the kid usually has a pretty frustrating early childhood, and then a pretty powerful experience the first time he or she meets the Deaf Community and is exposed to ASL.
In rural areas, there is often no Deaf Community, so they just make do. They’re usually taught English and have a very difficult time in school, and at home, just for expedience, the family ends up inventing signs for everyday use. These are just signs corresponding to English words (and usually only concrete nouns)—no grammar or anything like that.