This has always been a permissions issue for me. If memory serves correct (which it may not), that program gens the Pos Pay file from the server. In other words, on the App Server, go to Services find the Sage 100 service, and look at the account running it. That account should be a domain account that can Read/Write to the location of these other network shares you are trying to save to.

One way to check is to actually log into the App Server's desktop using the domain account from your service (IOW don't login as yourself but login as the service account). Then open a File Explorer (Windows Explorer) window and browse to the location of one of these network shares you're having trouble saving to (skip tsclient locations for now). Now just try to manually create a TXT file. If you can't browse to the location or can't create TXT file, that account doesn't have enough permissions.

If the service account uses LocalSystem then definitely that is a problem since by definition it cannot open a UNC path. With everyone out of the system, change it to a domain account that has not only access to the App Server but to the other network shares.