Honestly, if you're concerned about any effect on your credit score, you wouldn't go for another card with 'a decent interest rate', you would/should never carry a balance. That has an effect on your score.
I have no idea what the interest rates are on our cards, and I'm sure they're sky high. But I'll never pay a penny over what I owe each month-- to interest rates.
We use credit for everything and amass great perks/rewards/cashback. I remember hearing years ago that cards that offer rewards as such, don't typically have good interest rates. Almost an oxymoron!