Can you tell us why you've ruled out the bike you have? Maybe it's worn out, or you'd have to do expensive upgrades, or it just doesn't fit you well? Or have you decided that a road bike isn't the right tool because it's a road bike? (A road bike is perfectly suitable for true credit card touring.)
But to actually answer your question, almost any bike that fits you can be used for touring. Here are the priorities:
1. Fit, therefore comfort. I'm tempted to list this three times. Insist on comfort! A professional fitting might be money well spent.
2. A low enough low gear. It can be as the bike comes, or change things to get there. Often you can change just the freewheel or chainring. Low gear should be in the 20-25 inch-gear range, even for credit card touring.`
3. Cargo capacity, which should not be an issue for credit card touring. (A seat bag and handlebar bag should be enough.)
4. Sturdy wheels; for CC touring 32/wheel is enough. No "designer wheels", i.e. low spoke count, fancy patterns. If a spoke breaks on one of those, you won't be limping to a bike shop or fixing it in the field.
I'd go ride a bunch of bikes and get the one that's has low gears and at least 32 spokes per wheel, that's the most comfortable. (Hint, you may wish to upgrade the seat.)
5. Finally, don't automatically rule out recumbent bikes. They cost more and are harder to research and buy, but most of them make great touring bikes.