In terms of connectors, the HX 650W comes with a single 6+2 pin plug on each PCI-E cable, so you're fine.
I would be prepared to replace the PSU if necessary. You're probably
okay but the HX 650 was one of Corsair's cheaper units (only 80+ bronze rated) and by this point, it could be 9 years old. Depending on how many hours it has been in use, what load it has been subjected to, and how well cooled it has been, it'll either be totally fine, or unable to supply the ~25A on the 12V rail that the 1080Ti will peak at. If the PSU is too tired to handle it, you'll notice crashing at peak loads where the reference 1080Ti can draw up to 300W for short bursts (typical average gaming load is 250W).
Your PSU is rated for 38A loads, but PSUs do wear out over time, and as a rule of thumb you should expect them to be at around 100% load when the warranty runs out and lose 10% per year from that point onwards. YMMV and this is only a rule of thumb.
Thankfully you've won a Founder's edition that consumes power as I've mentioned above; It's one of the most power-supply friendly versions of a 1080Ti you can get. Common models usually include a small boost to clockspeeds and voltages - Something like the EVGA 1080Ti FTW3 is typical and that one consumes an average of 280W with peaks of 360W, whilst higher factory overclocks from Asus ROG and Aorus can average almost 350W and peak at well over 425W. I would almost expect
your old 650W PSU to fail with one of those monsters!