How can I obtain access to a cloud Instance(s) for testing?

I will need one or preferably more) instances for integration/system testing, perhaps including some load testing.

Will I need a paid instance, or are their resources available to use?

You can get a developer instance here, Sign up - Try Atlassian Cloud | Atlassian

For the other instances you can always use a small Jira instance. Like a 10 user free one. You can then always opt to get a paid version so you can do more customizations in Jira and thus possibly your app for testing.