No credit card required
Browse credit cards from a variety of issuers to see if there's a better card for you.
You do still have baddies in the form of monthly delinquencies.
I would contnue your GW campaign. Payment history is the most highly-weighted category.
With 6 open credti cards, presuming a couple are major bank cards, you have adequate revolving credit.
Increasing your credit wont help directly other than increase the denominator in your % util calculation. You can achieve the same % util without any credit limit increases by simply keeping the numerator low, ideally having all accounts but one report a $0 balance, and one reporting less than 10%.
Get your installment loan at least a year prior to when you plan to apply for a mortgage, thus giving you a year of history on the account, with the hard inquiry having dropped from scoring at the one year mark.