It sounds like the INQs that you listed are all from EQ. I would recommend getting a copy of all three of your CRs and see how many INQs you have. Remember - INQs that are older than 1 yr do not count against your fico scoring.
Also, unless you plan on doing BT, I would stay away from the Slate and instead get a Freedom and/or CSP. Freedom at least.
I don't know if the OP's chances of approval for a CSP is good considering her scores and the CLs on her other TLs. Maybe the Freedom.