I would like to be able to fill the form out and then send to a client for signature

Can I fill out the form in Jot and then send for signature to an external party with the fields retained?

Yes, it's possible to send the email to someone else just to review and sign it. 

If you will be sending forms to the same person always, then you could just add an email notification, this guide will help you doing so: https://www.jotform.com/help/25-Setting-up-Email-Notifications 

If you would be sending the email to a dynamically changed recipient, then you could just add an email autoresponder, this guide will help you to do it: https://www.jotform.com/help/26-Setting-up-an-Autoresponder-Email 

Now, you could leave the signature box to be always shown or you can just show it when the second user is reviewing the submission. 

If you decide to show the signature box just when the other user is reviewing the form, then you could use the Get Form Page URL widget and take advantage of conditions. 

How-to-Add-a-Widget-to-your-Form

This guide will  also help you to set up show/hide conditions: https://www.jotform.com/help/316-How-to-Show-or-Hide-Fields-Base-on-User-s-Answer 

You may set up a condition like this: 

If Get Form Page URL contains "edit" then show Signature box. 

The word "edit" will be present on the form's link as your user will be editing the submission.

I hope this helps.