This sounds like you took the steps you needed to take to attend the hearing and get your license back but that you never went through the rest of the administrative process to reinstate your license. Iowa is only obligated to send notice to your last known legal address. So if you moved to Missouri during that time, they may not have found you.
In any case, the only way to take care of this is to contact the DMV in Iowa and to see what hoops they are going to require you to jump through in order to clear your driving privileges so that they will release their hold and you can get a Missouri license. This may turn into as simple a matter as your just being able to corroborate that you did indeed have your hearing and your DUI was dismissed. It may be a matter of your having to pay reinstatement fees. This may also turn out to be a more complicated problem. I have no way of knowing the underlying facts that caused the hold. The only way you will be able to find out is directly through the DMV.
If this is a more complex matter, the fact that you received no notice may be relevant, and if what the DMV requires from you seems over the top, with interest charges and penalties beyond what you would have had to do and/or pay originally, you could retain counsel and appeal that based on lack of notice. As you are not in Iowa any more, you may want to contact an Iowa lawyer to handle the appeal for you.
Frankly, people in this situation usually find that they owe money and many tend to pay it rather than fight it, because it's the quickest way to get the matter behind you and to get your new license. But there is an appeal process for DMV decisions, and you'd be entitled to challenge a DMV order based on lack of notice or any other reason that is relevant to the hold.