Actually the animals have slightly different encumbrance rules. They can carry only as much as they weight themselves, while the character can carry significantly more. Also the animals have only 20 "item slots", much less than the character. Still, encumbered dog is very slow, though a single punt won't encumber it that much (a single raft though is enough to slow any dog to a crawl, assuming the dog can carry it at all, only about 50% of dogs can take it).
As for finding a punt, I suppose it is the same situation as I used to have frequently - the punt wasn't loaded after coming ashore, and still lies somewhere waiting for a lucky finder. Retrieving it would be a lot of work, probably requiring combing the forests near rivers and lakes recently crossed. And another punt/raft for it.
You CAN look into msglog.txt, jump to the end of file and start searching for a "punt" in reverse direction (the line will be like "(000000):jcic:[p]{046D019D} | punt", the hexadecimal number there is a coordinate of the tile), but it will work only if you have dropped, or entered the tile with it and looked, or something. Not all actions are logged.