I'm also betting the edge needs some touching up on hones, prior to stropping. As mentioned, paper-cutting should be easy from the stones/hones first. Work the edge to full apex (form a burr) on the hones, then stropping can be used to clean up the burr and refine the edge.
Another simple and inexpensive option, if you're so inclined (and if you don't yet have any other sharpening tools), is to 'strop' the edge on some medium or fine grit wet/dry sandpaper, laid over the top of your leather strop. Any grit from 400 and up should work. I'd suggest starting/practicing with higher grit, like 800 - 1000. That'll give enough grit (likely) to effect some noticeable change on the edge, but at the same time, it will minimize the damage if your technique is a bit off. It's the same exact technique as stropping; keep the pressure light and the angle conservatively low. Make just 2 or 3 passes at a time, and watch for the burr formation. When the burr forms along the full length of the edge, turn the blade to the other side and 'flip' the burr the other way. Use progressively lighter pressure to gently abrade the burr away as much as possible; keep verifying the edge is staying sharp while doing this. You obviously don't want to overdo the burr removal, if it's making the edge dull again.
Using the sandpaper on the strop will convex your edge a bit. Using a firmer/harder backing under the paper will minimize that, as will using lighter pressure also (always, always keep pressure light). I'd suggest praticing this technique on some cheap knives first, to get a feel for it.
Here's a good reference for stropping technique. It's posted as a 'sticky' at the top of the Maint forum page:
Stropping: angle plus pressure