Yes, it is true. Having active credit cards on your record is no problem. It shows that card issuers are willing to extend credit. If these cards have no balance on them, you should have no trouble getting a new card. Apply and start using it.
If there are balances on your other cards, it is a different issue. You should pay them off, or transfer them to the card with the lowest interest rate and pay that off as quickly as possible. Then make sure you never have a balance on a card that exceeds about 30 percent of the credit limit.
Paying the monthly charges in full each month is the best policy. It also means not paying any interest.
Some banks will allow you to upgrade your current card to something else. I have done this twice with no problem.