Is it legal for me to drive in one state with a DUI in another state?

That would be a no.

Did you attend court on the scheduled dates? If not, there may be a warrant out for you. Have a lawyer check on whether or not this is so, and make arrangements for you to make a court appearance and have the warrant vacated. You definitely need a capable lawyer who might be able to straighten things out. Do not just ignore this issue, because it will not die, and it will very likely come back sometime in the future to trouble you. If the court entered judgment against you on your non-appearance, it probably did suspend your licence. A suspension in one State is a suspension in every State. Therefore it is unwise for you to drive until after this is all straightened out. If you must drive, even if it is a bad idea, then be 100% certain to obey every single traffic law, so that you are not stopped by an officer, who will almost certainly check your record.

Nope, it's not legal. It's basically the cop's discretion whether to take you to jail or not the next time you get pulled over for a minor traffic violation. You should hire a lawyer to go to court to recall the warrant. A lawyer can do this without you present if the underlying charge is a misdemeanor, so it is less risky for you.

You are playing with fire and are probably not getting notices or mail because the information is not correct. Shortly you out of state license will be invalid, as will be your insurance and you will have a warrant out for your arrest.