To properly track a conversion from AdWords you will first want to determine a conversion page. Many times this will require you creating a "Thank You" page that will load after a contact form is submitted. You may need create this extra page and change your current contact form setup to submit and change to this page.
Applying the Code to the Conversion Page
Select your page that you would like your conversions tracked from.
Add the Embed Code widget to the lowest canvas on the page that is not a site canvas (avoid site canvases that show with diagonal lines)
Ensure that the widget is not overlapping any interactive objects with a link. The widget will be hidden but could cause linking issue if it overlaps a linkable object.
Paste your code from AdWords into the Custom Code field of the Embed Code widget.
Submit a Contact Form to Another Page
In order to support AdWords conversion tracking your contact forms will need to link to another page on submission instead of the default switch to a canvas view. Once you've created a Thank You page you will need to update the submission settings on any contact form on your site.
Select the Submit text (may be named similar) on you contact form canvas.
Go to the Click Action settings and change to link to Page: (select your page). Check the settings on any color button behind the text as well.
Be sure the Submit Form option is checked.
Preview the change and test your contact form.
Repeat for additional contact forms.