The plasticky type outer bag on my 344 needs to be trashed. Can someone recommend a replacement? Not picky about what it looks like, just that it works.
The best thing to do would be to go to a vacuum cleaner repair shop in your area, and ask for a replacement cloth outer bag for the Hoover Guardsman upright, which shares the same basic design concept as your 344. The bag you will likely get will be dark gray with a black herringbone pattern superimposed on it, and the Hoover roundel logo and the words "allergen filtration system" in the upper left-hand corner, in bright Orange lettering.
Below is a picture of that bag, as it appears on my own Hoover. It includes a plastic standpipe inside which allows it to be used with Hoover type A bags.
For a functional bag, try the ones with the Type-A filltube, you can use the Type-Y HEPA (apparently) bags in them, aswell as the paper Type-A bags of course...
This one on ebay should fit and work perfectly (after looking up what a Hoover 344 is on Google Images), it fits pretty much all Convertibles (US) and Seniors (UK):