Yup, agreed with the others - you want a stripping basket, and never mind what the small-minded by-standers might say about it.

There are a number of styles - some are like a plastic basin on a belt (see the Orvis one somebody else posted), and it's possible to DIY one without too much trouble (surf YouTube, lots of options). There are also a few mesh models (like THIS) and a foam one, which is my personal favorite - if you can find one.

The cones or spines inside are an attempt to keep line from coiling and getting knotted. They work, sorta. Definitely better than what you're dealing with now, but they do take a bit of practice to get used to it - the trick is to make sure that you always strip into the basket. That sounds hideously simple, but it's harder than it seems once you get out there.

If you're always standing on the shore and aren't wading, another possibility might be a collapsible leaf basket, something like THIS. Those aren't ideal since they're floppy in the wind and often too big or too small or not tall enough, and they often aren't as easy to carry. But they are cheap, and you can buy them at lots of places - not just from specialized fishing shops. Bring a big ziplock bag and a towel; fill the ziplock with rocks or sand, then put it in the bottom to keep the thing from blowing away. Wet the towel and arrange it on the bottom so your line doesn't snag on the rock bag. Other variations on that theme will work as well.

Good luck!