The advice on the forums is usually to close Credit One and First Premier when possible. I don't have any experience with either one, but if you search the forums you will see those are predatory cards that you want to get rid of as quickly as possible, so that would be an excellent place to start.
You have other cards with benefits (i.e, cash back from Quicksilver, rotating categories with Discover), so Credit One and First Premier are really just dead weight.
If you list information such as how long you have had the cards, others may be able to help as well. For example, my first thought is - with your Discover FICO score and a good history with the Walmart card, to call Credit Solutions and request an increase.
If your Capital One cards are eligible, you may also want to consider an account combination. This involves combining the two accounts into one in order to create a single account with a higher credit limit. I would recommend requesting a CLI on each card before combining them into one if they are eligible for a CLI, however.
If you decide not to do an account combination, I would recommend PC'ing the platinum card to a Quicksilver card. You can request this product change (PC) via chat within your Capital One online account.
Closing the two accounts listed above will not affect your AAoA for 10 years, so it would be beneficial to close them. It also should not change your FICO score significantly since your overall utilization does not appear high.
Individual utilization appears high on the Credit One and First Premier cards, however, so this is yet another reason to pay them off and then close them.
How recent are your collections and late car payments? This will also impact what is possible with your exisiting cards.
If your negatives are not too recent, I'd recommend applying for the Barclay Rewards card based on the TU FICO score you have listed, as I believe your chances for success in getting the card would be solid.
Best of luck! 
