Balance Transfers

My wife and I have two credit cards with a balance. One has an interest rate of 11% while the other one is soaring to 22.99%. We used these cards to make ends meet during a 6-month layoff, so the balances are $8000 and $6000 respectively. I believe that the card with the highest interest rate has such because we were late on a couple of payments. The late payments were due to the fact that each of us “assumed” the other had paid the payment. We have decided to use an automatic payment option through our bank to make sure that doesn’t happen again.
We have been told via the media that it is better to transfer balances from high interest rate cards to one that has a low rate. But the problem I am having is that I can’t get a credit card from anyone else. I tried Simmons First and Discover Card. Discover has a 0% interest rate for 12 months. Both card companies’s declined my application stating that I already have too many credit cards out with a large balance. And, every time I apply and get declined, that also goes against me. My credit score is great so bad credit isn’t the issue.
So my question to the experts is: what should I do here? How do I get a low interest rate card when I keep getting told “no?” Are there good alternatives I can explore? I’m not really excited about getting a second mortgage for this, but is that an option that should be explored? Your advice?