I’m really scared because I haven’t been able to book a seat reservation from Paris to Amsterdam on the 8th of July. Does this mean we have to buy a full fare ticket?

Yes  or travel on a other Route 

For Example via Lille or via Karlsruhe 

Easiest Route would be Amsterdam  - (Change in Belgium) - Lille - Paris 

I have made a reservation for the Thalys from Paris to The Netherlands by calling NS International. You can reach them via this phone number: +31 30 23 000 23.

Maybe it's good to know that the price of a seat reservation is rather high, namely €30,-. Furthermore, they said on each train only a few seats are available for Interrailers. If they're sold out, you can not take the train anymore. 

An alternative would be to go from Paris to Lille first, which takes a bit more than an hour. There are more frequent connections on that line, so you might have more luck. I think it's best to call the SNCF for that trip. I paid €10,- for a seat on a TGV-train. If you can find a (cheap) ticket from Paris to Brussels at once, that's also fine. 

From Lille, you can take a combination of Regional and Intercity-trains to Brussels (about 1:45 hours, 1 interchange). I figure these reservation costs will be low, if they're even necessary at all. 

From Brussels to Amsterdam, there is an hourly Intercity Direct service. That takes roughly 2,5 hours. I believe no reservation or fee is required there. 

You should be able to call NS and check if there are seats left.  No booking fees, they do accept VISA. Don't forget the country code (+31)