I'd just pop them if you haven't already. I'm not sure if they carry over because I popped or traded them, but yeah, as someone said, they aren't really the keys to get through. I only had earned two old souls, but because I had 1 million soul memory, I was able to progress past the Shrine of Winter in my original playthrough. Its likely not solely dependent on you just having the souls in your inventory as much as it it just having a certain amount of souls racked up. And anyways, you'll just end up getting them back on your second playthrough regardless, so no sense in keeping them unless you're planning on not leveling up on your second runthrough. As a side note about NG+ in general though, I don't understand why everyone talks about there being an immense difficulty curve from the original playthrough to NG+. If anything, I've been having an easier time and been progressing at a much faster rate. Something that took me a couple of days to do in my original playthrough was done in one sitting on my second playthrough. No-Mans Wharf was still absolute hell for me though, just less so.

I read somewhere that all "key" items are removed from your inventory.

It's not the actual souls that gets you through the door, it's defeating the bosses. They just play it out that way since you get their souls by beating them anyway. The souls in your inventory are only used to forge weapons/equipment. They e do carry over, but you won't be able to open the Shrine Winter with those alone. You can consume those souls the moment you get them in your first playthrough, so if they were the key to the door then you wouldn't be able to progress through the game.