TGV and OuiGo trains

You will probably be OK. People travelling for work often travel on a few hours notice. But, they pay for the privilege. Tickets bought on the day cost painfully more than those bought weeks in advance. You pay what the market can bear.

As Chris said, trains very rarely sell out, but buying tickets last-minute will cost you a lot of money (probably three digits one-way).

Ouigo might even have a requirement to buy tickets on-line. They have no "Flexible" price listed on the TGV website. It says, "I stand on the dock at the latest 30 minutes before the train departs." I see they want an extra 2 EUR to plug your computer/phone charger in. Bring your bottled water on board, they sell nothing. But the price can't be beat. For tomorrow, the price is 55 EUR, the cheapest regular TGV is 86 EUR. A month from tomorrow, it is 19 EUR and the cheapest nonrefundable TGV Prem's fare is 45 EUR. At 19 EUR it might be smart just to buy it on spec. If you miss it, you're only out 19 EUR.

I've taken that train both ways, and what stands out is how many cars there were on the train. And while they were crowded, they were nowhere near fully booked. So unless it's a holiday weekend, I would assume you'd be fine.

Look on the calendar for the "flexible fares" - that's where you'll find the best deals.
If you go mid-day to early afternoon, you will be sunk.
Better take the chance and go early.

Strongly advise you not to wait until the last minute. Our French trains are increasingly crowded, and last-minute fares can be exorbitant. Have a great time! Faith McLellan

If you can get a OuiGo train, buy it now. The tickets are dirt cheap, starting at €10, so it's no big deal if you miss it. I would be conservative about making sure there is plenty of time to make your connection. And you need to be at the OuiGo check-in 30 minutes before departure.