Can I check the new version on a couple of test site collections only and then trigger the New Upgrade for the rest of the site collections in my tenancy?

No, it is not possible. Once you click Upgrade Now at any of the site collections, it will automatically trigger the upgrade process for all of them.

You may test the sites using the Sandbox Mode before upgrading. 

To check the new design elements and connections you will be able to edit page, test them and preview using the Preview button for those elements or sections. Using the Sandbox Mode you will also be able to check if there are new features in Theme Builder.

Tip:
If you need to check more sites in Sandbox Mode before upgrading, do not close the Sandbox Mode just yet. Upgrading from any of the sites will close the Sandbox Mode from all of them

Important:
You cannot save the changes: insert or update the Design Elements and republish Theme Builder customizations.

Also, you might have the test environments for such purpose and check the new version there.

Alternatively, you may preview any of the versions starting from ShortPoint SPFx 7.1.5.xx, using the sp-preview=<version> query string at the end of your URL. Please, check this solution article for more details: Preview Any ShortPoint SPFx Version on a Page Using a URL Parameter. 

Related articles:

• Preview Any ShortPoint SPFx Version on a Page Using a URL Parameter
• I don't want to upgrade my whole tenant. Can I enable New Upgrade for some site collections only?
• Can I Preview any of the versions you released since New Upgrade?

Main articles about New Upgrade:

• ShortPoint SPFx New Upgrade for Office 365 and SharePoint 2019
• ShortPoint SPFx Auto Upgrade Feature
• ShortPoint SPFx Sandbox Mode Feature

You may check other FAQs about ShortPoint Upgrade in this folder.