How do I format {{$timestamp}} as MM/DD/YYYY in Postman?

You could use moment.js with Postman to give you that timestamp format.

You can add this to the pre-request script:

const moment = require('moment');
pm.globals.set("today", moment().format("MM/DD/YYYY"));

Then reference {{today}} where ever you need it.

If you add this to the Collection Level Pre-request Script, it will be run for each request in the Collection. Rather than needing to add it to all the requests individually.

For more information about using moment in Postman, I wrote a short blog post: https://dannydainton.com/2018/05/21/hold-on-wait-a-moment/

Use Pre-request script tab to write javascript to get and save the date into a variable:

const dateNow= new Date();
pm.environment.set('currentDate', dateNow.toISOString());

and then use it in the request body as follows:

"currentDate": "{{currentDate}}"

My solution is similar to Payam's, except I am using

//older code
//postman.setGlobalVariable("currentDate", new Date().toLocaleDateString());
pm.globals.set("currentDate", new Date().toLocaleDateString());

If you hit the "3 dots" on the folder and click "Edit"

Then set Pre-Request Scripts for the all calls, so the global variable is always available.