When you refer to the Aramaic of the targumim and rabbinic literature, be aware that you are referring to a number of different languages! Linguists refer to the language of the Yerushalmi, for example, as Palestinian Jewish Aramaic and the language of the Bavli as Babylonian Talmudic Aramaic. Some of the targumim belong to the former group, while some have elements of the latter group in them as well. Note also that the language of the Zohar is different again, so passages that you are looking at in your siddur might deviate grammatically from what you see in, for example, Shas.
In my opinion, just as you are not looking for Christian sites, you're also not looking for Jewish sites. You want proper, academic sites that don't support claims with reference to midrash or hagiography. Your intention, if I am not misunderstanding you, is to learn Aramaic in order to improve your grasp of texts that are written in that language. Lots of universities around the world offer courses in various Aramaic dialects, and some of those universities will offer classes online. None of them will be free.
I hate to say it, but I think that your best bet is printed material, not online courses, and I think that the way you were going about it originally was very good. Start with texts that you want to be able to read, have an English translation with you while you are reading them, and consult a dictionary. English translations of all of these texts exist online, if that's your preference.
At the same time, work slowly (or at your own pace) through an introductory grammar, of which there are several. You can find some suggestions in the answers to this question (one of which is by me).
By the way: while Hebrew and Aramaic are related, their relationship is not quite that of "sister languages". It's similar in some ways to the relationship between English and French: in other words, they belong to the same language family, but to two different classes within that family. English is Germanic and French is Romance. Similarly, Hebrew belongs to the Canaanite group, which is distinct from the Aramaic group of Semitic languages. See here.