Reddit mentions of EMS USB2 Sharp Shooter - Support up to Two PS/PS2 Joypads or Dance Mats on PC or Guncon Compatible Guns on PS2.

I always had good results with this.
https://www.amazon.com/EMS-USB2-Sharp-Shooter-Compatible-PS3/dp/B008FG72X4

I recently had some experience dealing with the same adapter you're talking about. I didn't need to install xpadder or any drivers to get it to work, but mine had the axis problem where you couldn't press opposite directions simultaneously (later models could fix this by pressing start+select+up for 3 seconds).

My recommendation is to uninstall xpadder and see if you can get it to work without it. If the adapter came with drivers maybe try those but I wouldn't have too much faith in them. If that fails, get a better adapter. Currently using this, works great with one pad, haven't tried with two, you may need a different adapter for two as this one is designed to use a light gun on the second port. You will need to install the drivers from the website, they're old, but they still work. Also, a whole list of recommended adapters is here

I realized this is actually a problem with the specific PS1 to USB adapter that I have. It's incapable of delivering both arrows (right & left) at once, and the method I had of mapping 126, 128 to push both arrows won't work since joy2key can't map raw data.

The solution, in case anyone else has this problem, is to just buy a different adapter. I found a list here, and this is the one I bought.

The point of the N64 C-buttons was to provide more directional or camera inputs. For most games that aren't Mortal Kombat Trilogy, mapping the C-buttons to the right analog stick X and Y is more than sufficient.

I have three sets of controllers now:

• an EMS Dual-Shooter adapter (2 Playstation gameports for 1 USB 1.1 plug) with either two Street Fighter 15th Anniversary Edition gamepads (Saturn/Genesis) or ArcadeMX two-player joystick (MAME/Supermodel) plugged into both ports.
  
• 4 wiimotes with bluetooth (Wii and NES)
  
• 2 Playstation 4 gamepads, and 2 Zhidong V gamepads (everything else)
  
  Configure everything once, and you're golden.
  
  EDIT: Thanks to /u/itsamamaluigi for the correction. I meant right analog stick. :)