I am looking to essentially promote a subsite up to its own site collection on a different content database, but all data and currently running workflows need to be preserved. Currently, I have a SharePoint 2013 environment with a single web application, single site collection, and then several subsites. I now need to move these subsites to their own site collection, but I have several lists with 2013 designer workflows currently running. The status/stage of these workflows need to be preserved. Is this possible? I have seen a few people mention backup/restore and then delete what isn't needed. But does this preserve running workflows or does this only copy workflows over to the new site collection?
As you are running sharepoint 2013 workflows their status will be not preserved. SP 2013 WF is decoupled from Shaerpoint and it is separate product. You could try a scope restore procedure (REF:https://channel9.msdn.com/Events/SharePoint-Conference/2014/SPC356)
In my opinion best option you have is to migrate the site and then re-create the status of the workflows (you might need to rewrite them a bit to allow for this kind of functionality)