Dear Donna, It is very possible your source is correct and you have Zoysia grass. You should confirm this for sure with your local extension office or your favorite garden center. Zoysia is a warm season grass, very adaptable to your area. Grasses with rhizomes are difficult to control as the root system can be extensive and robust. A physical barrier that is deeper than the root system is only practical for small areas. A soil/turf open edging that is 4 or more inches deep will definitely slow the spread and also gives you an easy edge to apply a herbicide like you have been using as runners emerge. For Roundup to work the best, the grass will need to be actively growing and as you have already experienced, will likely need to be applied several times a year. Roundup can be applied using a “wick” type of applicator and does not need to be sprayed limiting drift issues.
For native plantings, contact your closest arboretum for help in making selection choices. Native plants will also promote populations of native insects as well as attracting native wildlife. In recent years, retail nurseries and garden centers have also seen the benefit of offering natives to their customers. You may have to travel a greater distance to get what you want, but worth it. A quick web search showed several native plant societies and potential sources of native plants in your area and into Missouri.
Continued good luck with your gardening.
Sincerely, Richard Hentschel