No credit card required
Browse credit cards from a variety of issuers to see if there's a better card for you.
Congrats on your engagement! Here is my 2 cents regarding your medical bills- If you have any kind of documentation that you had state medical coverage, you can use that to dispute the bills. Their failure to properly bill the state should result in removal. The bills cannot be filed now because the statute for filing is usually 12 months to 2 years, depending on the carrier. The flip side is like you stated, wait it out to be removed. Since you have other negative items on your report, you will only gain 10-20 points max for the medical bill removal. The biggest jump in score comes when your last negative item is removed.
As far as what to do next, you wait lol I would not apply for anything else yet. Which credit card did your fiancee add you to? Do you live together? Not all companies will report an AU account if you don't share the same address. Keep in mind that his utilization will affect your scores too. Use your cap 1 card and pay all but a small amount each month so it will report and you can build credit. You can get increases to your limit as you establish history.
Should you apply for other credit right now, your limits would be minimal - if you actually were approved. It is better to wait a few months for everything to start reporting and your scores to establish before applying for new credit. Credit building is a marathon, not a race. You are getting started in the right direction!!!