Hi Allison,
You can deploy a new copy of a solution by changing the settings of the deployment.
In the Top right corner, the three parallel lines (looks a bit like a hamburger) is a drop down where you can change the settings. If you uncheck "Use Existing Items in the Organization", it will redeploy everything for you.
If you then need to use some of the existing services (reference layers or other data), you can then just switch them in on the webmap.
Andy Shoemaker