I'm not an expert, but I play one on TV...
We tried for years to just go ahead and kill debt but couldn't get on top of it. We put so much money into beating VISA and lines of credit, and stuff, that when anything at all went wrong, our whole house of cards crumbled, and we'd feel forced to reach for credit, putting us back in debt by whatever we had just paid off. It was a bad cycle.
What did make a difference was a 2-pronged approach to beating the cycle.
First key element - a strong and reasonable budget.
By budgeting more effectively, we were able to continue making our base payments on the essentials and see the spots where we could save. Holding to the budget while paying things down was key. And in our budget, we allowed cash for the unexpected. Things like replacing cars, major home repairs, and so on.
Second Key element - Building up a "War Chest" or Emergency Fund
Once we had a couple months of effective budgeting, under our belts, our family got serious about saving. We set a goal to have a war chest of $15000 in order to stave off whatever emergency we came across. Knowing exactly what the budget looked like meant we knew where it was safe to cut, and we had a good handle on exactly what debt we carried, where, and how much it was costing us month over month. This meant we could maintain minimum payments and even get ahead a little while building savings. And we did - in the end we settled for a $10,000 war chest in a TFSA (Think Roth, but in Canada).
There is a lot more to your payments than just interest, so focusing on the highest interest account may not work for you. In our case, I took on a side gig and pushed more money into the debts from that. Any gifts, windfalls and tax returns went straight to debt. The shared approach of budgeting and declaring war on the debts meant we went from owing lotsa cash to debt free in a matter of months - definitely under a year. In that time we also bought 2 cars, put on a new roof, and went on a spendypants Caribbean vacation.
It can be done.... Our paydown is all documented in my (lifestyle) journal, but there are pages and pages of it, so you might get tired of reading.