Algae spores are nearly everywhere in non-desert environments. They can even travel in the atmosphere where they enter your water supply directly, or indirectly when they get incorporated into raindrops.

I keep a kiddie wading pool in our backyard for our dogs. I fill it with chlorinated city water. Within a week, it is exhibiting algal growth. IMO, it is impossible for you to exclude the algae from your pond ecosystems.

Even the soils can hold dormant algal spores. I don't think running your water from one pond to the other will "filter" out your algae. In fact, it might even increase it after the initial decrease.

You can physically rake out the green, string algae. (Called "filamentous algae".) You need to deposit it away from the watershed that fills your ponds - or the nutrients will run back into your pond water after the algae decays.

Algae does require sunlight, so deepening your ponds would be one option that could add much more water to your total system without adding much more algal growth.

Another option would be to use a copper-based algaecide. A lot of our pond management experts use Cutrine Plus. It is used in ponds with fish and humans swimming. In the CORRECT dosage and treatment, it will kill all or most of the algae, but not harm the larger fish. I believe it may be toxic to some of your amphibians and zooplankton in higher doses, but with perfect micro-dosing some or all may survive?

If your ponds are mostly support for terrestrial wildlife, then I believe the treated water would be safe for them. (The product label says it is safe for watering livestock when used according to the instructions.)

OTOH, the most common types of algae make the water look and smell unsightly, but it is still safe to drink. (Some types of blue-green algae are toxic.) You might just let your wildlife deal with the algae. (I have one dog that prefers to drink from the kiddie pool when it is green, even though there is a bowl of fresh water right beside it.)

What are the main types of wildlife you wish to support? That might get you the most useful responses to your post.

Good luck on your mini ponds project!