...a budget lens I can just keep on the cam and use for anything.
What kinds of "anything" are you anticipating might come up? As the saying goes "jack of all trades, master of none"... and a very general purpose lens like the 18-200mm doesn't have as great image quality (IQ) as other lenses that are a bit more specialized.
Which camera/body did you get? and which 18-55mm? (the latest is actually a very good starter lens).
"action scenes" as in sports? indoors? or outdoors with plenty of light?
Going from the 18-55mm to the 18-200mm will be a step down in quality at comparable focal lengths, and unless you're getting the 18-200mm for really cheap, I think the 55-250mm might be a better choice even if it means its second lens. Amazon does promotions often where they discount the 55-250mm heavily with camera purchase, so the second-hand market might have some great prices.
Also, as you mentioned, there are non-Canon "superzooms" to consider. At the beginner/non-professional level, they can be a good value too.
I haven't owned the Canon 18-200 but I have owned both the Sigma 18-200 & the 18-200 OS (same as IS in Canon lingo) plus the Tamron 18-270 VC (same as IS) and I got very good images from them, and my daughter still gets excellent images from the Sigma 18-200 NON IS style lens. The range they cover in a single multi purpose lens is great, but they are a compromise between versatility & absolute Image Quality (IQ) and some demand IQ over anything, even when they themselves don't benefit from it. Unless you intend to print LARGE, or view the files on screen after heavily cropping them a superzoom will do quite well. There are instances when you'll notice it's flaws, but in general they will be far fewer than most suggest. You'll have many more opportunities with a 18-200 and generally speaking when using it hand held the IQ will depend on your skills (high enough shutter speed, rock solid stance etc) to get the most from it.
People are all about the IQ a lens can deliver but they seem to forget that more often than not it can't deliver that extra bit of IQ unless everything that influences the photo was done perfectly. Hand holding & slower shutter speeds almost always cancels out any differences between a perfect lens & a very good lens in real world use. IF you get serious about photography as a hobby & decide you need to get better gear as your technique improves you may see the need for better lenses which deliver higher IQ when used correctly (usually on a tripod or monopod) then you can start analyzing the pro's & con's re cost / quality.