The Global knives must be able to keep and hold an edge to some degree because Cat Cora states she uses a Global santuko. However, in the episodes of "The Iron Chef" I have not as yet seen her actually use one. (I will admit that viewing that show is sporadic.)
If you google "knife sharpening" numerous references can be discovered. It sounds as if you need something simple and repeatable to repair those knives. Without seeing them it's hard to recommend which method might be of help.
There must be a kitchen supply store in your area. If there's a "knife guy" at that store, let him examine the problem and recommend a system for the skill-set you have. In that manner you will be assured that you can use and hone the knives you own anytime you wish. If you're worrying about your edges you don't enjoy using them.