You don't need to activate your pass to book reservations. It's best do deactivate again, until you start travelling.

30/5 is right after the long Whitsun weekend, so lots of people are travelling. There is usually more availability from London to Lille; in Lille you can take a domestic TGV to Paris. Or explore Lille: it's worth a visit.

The best place to check Eurostar availability is:

https://www.b-europe.com/EN/Booking/Pass#TravelWish

The best place to check availability of domestic TGVs in France is:

https://travel.b-europe.com/Eurail-GE/en/booking-tgv#TravelWish

Both web pages will only show train with availability.

Eurostar (and Thalys) trains have a limited quota which can sell out weeks in advance.

This is the case here, esp. after a long holiday weekend. No trains available at all on 30th May, even to Lille or Brussels. There are some on 31st May. It is a true bottleneck, the worst of Europe imho as there are no other railway options. Even regular 250€ tickets are almost sold out.

It is unfortunate that Eurail doesn't show availability until you've bought a pass.

Few alternatives :

https://www.seat61.com/trains-and-routes/london-to-paris-by-ferry.htm

or Flixbus (170€, not worth it at all)

I'd definitely take the Dover - Calais ferry or bite the bullet for a regular ticket…

- London St. Pancras - Dover Priory 10:07 - 11:12

- taxi (7£) or 30 min walk

- Dover - Calais 13:40 - 16:10 30£ (boarding 90 min prior to departure)

- free bus to the train station (every 15-20 min)

- TER Calais-Ville - Paris-Nord 17:38 - 20:56 reservation-free

It would be a fantastic adventure Let me know for questions

If you have a 1st class pass, then there are some more possibilities. Here's availability to Lille:

Also more availability to Brussels in 1st class :

- London - Brussels 06:16 - 09:12 38€

And then either another expensive train :

- Brussels - Paris 1h22 few trains available 32€

Or :

- Brussels - Tournai 10:13 - 11:15

- Tournai - Lille-Flandres 11:22 - 11:52

- TGV Lille-Europe - Paris-Nord 13:03 - 14:32 10€

You could stop for a few hours in either Brussels or Lille (only need to decide in advance).

Or last option which is reservation-free :

- spend a few hours in Brussels

- Brussels - Charleroi-Central 13:15 - 14:07

- Charleroi-Central - Maubeuge 14:24 - 15:20

- Maubeuge - Paris-Nord 15:42 - 17:41

You could swap passes for 1st class ones for a small fee.

In any case you need to choose whichever option you prefer ASAP.