Here are my thoughts.
NEGATIVE INFORMATION. You have a major negative that is scheduled to drop off in just a few months. (The charge-off at TU, which is why your TU score is so much lower than your others.). Great news! Your only other negative is a late (student loan) which will fall off next spring. Does the late appear on all three credit reports? When your negatives are all gone, your scores will be significantly higher. Am I right that these are your only two negatives?
CREDIT CARDS. You say that your credit card balances are all zero, and then go on by way of explanation to say that you seldom carry a balance. There's a crucial difference between reporting a balance and carrying a balance. It's a very important distinction for managing your credit cards. You will get a substantial score boost if you do two things:
(1) Make sure that exactly one card is reporting a small balance (e.g. $20).
(2) Make sure that the other cards are reporting a $0 balance.
That will cause you to have a very low utilization and it will cause you to get a lot of extra points. Note that you will get penalized if your credit report shows ALL ypour cards at $0. Let me know if you need help with understanding the difference between reporting and carrying balances. BTW, it is a really good habit financially to get in the habit of never carrying a balance. That just means always paying the full amount on the statement (after it prints).
The strategy outlined above (all cards zero except one with a small balance) is only important when you need extra points -- like applying for a new credit card or a loan. In general it's easiest if you use all cards naturally (though certainly keeping a close eye on spending). It's actually good to use each card at least a small amount, though it doesn't have to be be every month. Even when you are in your general use phase where you are not scrambling for extra points, it's still reasonable to keep your utilization down (less than 50%) and to pay them in full!
You should not apply for a new card right now. Wait until the charge-off drops off and until you get your credit card balances the way I describe above (one zero, one small balance). Then make sure that your credit reports conform the new state of affirs (on both counts). Once your credit reports confirm, then you should have a significantly better score and will be in a great place to execute your next step which was to apply for that third card.
The way you describe that third card, it sounded like you were going to try to open it as a joint card in both your names. Is that right? Many credit card companies no longer do that (though some do). There's a lot to be said for each person having their own cards in their own name -- it makes building credit easier and clearer to the CRAs. The joint card idea is not a dealbreaker -- you can do it if you want. It just may be better to have your own separate cards (but of course manage them together).
Let me know whether this all makes sense. This should be the first stage oif your credit improvement plan. Depending on how high your scores get after the late falls off, you may not need to do anything tricky with the Share Scure loan. But you should do this all first to see what you can achieve. A car and then a house are very real possibilities if you do things right.
PS. You didn't manage what tools you are using to check your scores and your reports. Can you clarify?
