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I have a very thin profile and only 3 year age of credit, I was approved with a SL of $7000. (Credit cards are 13 months old. Old paid off car loan bumps my age to 3 years)
In my profile is a current car loan, Target card and a card from my bank. Scores 660-680. You can see more of my info here. http://ficoforums.myfico.com/t5/Credit-Card-Approvals/Discover-Approval-7000-CL/m-p/4967159
You want to pay it down not pay it off entirely... that said..
Give Discover a whirl... or Amex .... both tend to be amicable to thin files.... Discover may not give you a high SL out the gate but it will eventually grow...
Ditto for Amex.....
=)
I'd suggest joining Chase Credit Journey -- it's a free credit monitoring website by Chase and provides a useless FAKO score but they do offer pre-qualification links on there. I charged off chase for 4 figures in 2008 or 2009 and expected to never be let back in. I joined Chase Credit Journey earlier this year, and when my TU jumped over 635, I got a prequalification link on that site. Clicked and approved, even with a hefty charge-off within the past 8-9 years.