I am writing my first letter to my oldest, closest German Schwabisch relative. We have never met or corresponded. My brother visited years ago. I am trying to reconnect with my father’s family in his hometown. I got her mailing address from a third party. I grew up in a mixed nationality American household so my German is 2nd hand at best let alone the elements of a personal letter. I don’t dare write it in English because I don’t get the impression that English is a part of her world. It also just seems rude. Please let me know what German/Schwabisch elements/phrases are considered appropriate. Also how do you confer respect for an elder while being warm. I want to build a relationship. I do not want to offend. Thank you in advance for your help.
There is no need to get creative here:
Liebe (Großtante) <her first name>,
<content of the letter>
<some good wishes>
 Optional but has the advantage of clarifying your relationship immediately, needs to be adjusted to the actual relationship. I actually called my great-aunts "Tante <name>" and I think that's pretty common. You could switch to that in later letters (if she is your great-aunt).
I'd advise against using dialect here. You wouldn't get it right. (I wouldn't either.)
I have a similar background and like to send postcards to my relative.
I generally open with „Liebe(r) [Name],“ as it is a pretty standard opening for personal letters. You can use it for friends and even acquaintances. It’s analogous to “Dear [Name],” in English; it’s not used in professional settings, but it adds enough decorum that it doesn’t feel too informal. The most recent letter I received from my German relative says “Hallo liebe/r [eurieka]!“ (*gender hidden) and we’re relatively close, I’d say. This demonstrates a bit more proximity than what you’re after, in my opinion. So sticking with “Liebe” is probably best here.
Typically I sign off with „Liebe Grüße“ (or even simply with „LG”). There’s probably other ways to sign off, but it’s warm without sacrificing propriety. I even had a very kind professor who signed with “LG” when she sent class emails.