Thank you for your post. I apologize for the late reply.
I need some clarification to help you. Is your FileMaker Server on your home network?
You have isolated the issue to that new router, so I would focus on that. Have you verified port forwarding has been configured on your new router? Are there any features included on this router that might block port 5003? Have you tried the checking the router's port configuration and rebooting it as well?
I realize it's been some time since your post, so if you have found the cause, please post here so that others having a similar issue can benefit.