Is your SS# on the Southwest card under your dad's account? If not, have him add it.
-Since I was in school last year and will be working full-time next year, should I fill out the 4506-T or just wait and re-apply for the Zync? (My income tax last year will reflect a significantly different income than I will have this coming year [13k vs. >100k, which was on my Zync application])
You put 100k on your Zync app and it's currently 13k? That will be a problem.
-My dad has offered to cosign on my first card since I'm being declined on my own. What cards could i apply for that allow this? Could I be added to make one of his existing cards a joint account?
Start by doing it with the Southwest card.
-Finally, other than getting a department store card asap to start building history, what are my other options?
How long ago were you in school? Maybe apply for a credit card for college students (see homepage for list). The credit line won't be fabulous but at least it will be an account. Whatever you do, stay away from store credit cards.
Disclosure: I am a moderator/paid staff of this site, which does have advertising relationships with some credit cards that are discussed and linked to. Regardless, anything I say is my honest opinion.
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