You have to be careful what you are tying equipment to yourself and/or the boat. Provides a good chance to become entangled and drown.
As long as any part of the boat is above water you stay with it (tidal waters).
If in a river don't go off the waterfall holding onto the canoe.
In a capsized boat it's likely you will only have your PFD, and whatever else you may be carrying on your belt. Assume a knife and a PSK in best case.
The gear tethered to the boat is gone with the boat.
You don't use dummy cords or tethers to yourself due to entanglement.
Example, you tether yourself to a day pack. But you leave slack so you can move around within the boat/dinghy/canoe. The loop of the tether gets caught on the seat, a cleat, or any other gear while the boat is capsizing...Not Good. Even if you are clear of the boat, you now have a water laden pack tethered to you, pulling you, and could get entangled on a sunken tree, rock, etc. and you drown.
back to topic:
So you make it to shore, assume a river, as stated it's only a little above freezing. Which probably means the water took you well down the path to hypothermia. You are cold and shivvering, fingers numb.
FIRE FIRE FIRE FIRE. It is all that matters at this point.
Clap your hands together, do some jumping jacks, push upsd, just enough to get the blood flowing then it's all about FIRE.
Don't waste valuable time and energy bringing firewood to you, go to where there is the most wood available and get the Fire built, pronto!!
Make the fire large enough so you can undress fully and dry the clothes out fully before nightfall.
Only after getting entirely dry and warm should you turn to shelter building. You can survive next to a hot fire all night, but maybe not in a shelter soaked and shivvering. Once dry, you'll probably be occupied with getting enough firewood for the night that you won't be building yourself a 3 bedroom rancher with walk-out bsmnt for a shelter.
If the scenario occurred early in the day you have time to dry out, fire, and build shelter, but, that is assuming a lot in your favor, better to plan for less time and more urgency than thinking you'll have all day long.
Don't jump to how to survive the entire night, when hypothermia could set in well before. Stay focussed on the thing that is gonna get you first, once it has been controlled, then move to the next thing.
Sorry about not mentioning the knife, but, except for maybe wittling a fuzz stick or two to get the fire started it's immaterial, you will use what you have.
I would be gald to have a $10 Mora at that point, anything above and beyond is gravy.