For cleaning up pee, use “simple solution”. There’s a carpet and hard floor one. They have it at Pets Corner, prob Pets at Home too. It’s enzymatic, gets rid of the smell and stain. I also use for vomit - dogs or kids!
I’m slightly shocked that you are talking about rehoming (albeit as a last resort) after two days.
Puppies have no concept that peeing inside is wrong. So, why wouldn’t they go outside, have a sniff and a play, come inside for a pee? After all, it’s what you do. Admittedly not on the rug or the sofa.
The thing is, persistence. Take pup outside every half hour or so, if he pees, yay! Lots of treats, praise. When he’s inside, watch him. The moment he circles, scoop him up and take him out. If you catch him in the act, no telling off (otherwise he’ll just go behind the sofa where you can’t see him), just scoop him up and rush him out. Even if it means he trails pee across the floor and up your arm. With a pup, you need to accept, the first few weeks are spent doing this. If you don’t want puddles, life does not go on as before. You have to be a full time puppy watcher.
At night, if you want sleep, have him in your room. He’s a baby and missing mum and his brothers and sisters. Dogs are sociable animals by nature. Take him out last thing for peeing. Set your alarm for say 2am or 3am. Carry him outside to pee.
It’s worse than a new baby. It really is. But the sleepless nights is over much more quickly.