Yes TGVs are very popular and can sell out days in advance, especially those heading south and to Germany. If you’d like to travel this summer, I wouldn’t wait long before booking. Have a look at this page : https://travel.b-europe.com/Eurail-GE/en/booking-tgv#TravelWish
Seat reservations : there’s a 10€ limited fare and then it costs 20€ until the train is sold out. They reduce flexibility (by a lot) but at least you know that you’ve got a seat.
Most French trains are crowded honestly as they don’t run often…