Best way to collect email sign ups on my website and use them in Flow

It looks like this site is not a SharePoint based site, am I correct? if so then PowerApps is not really an option as licensing becomes as issue.

I'm assuming this is why you have used Forms, because it can be embedded into any site and that you can easily automate with flow. Unfortunately there is not much control over what the form looks like when embedding. If you want it to look like what you have described, you may need a slightly more technical solution. As in create the page yourself using html and a backend language to call a HTTP Webhook in flow. But this requires a little more work.