I use an ergonomic mouse from a brand called Evoluent, which I have loyally used for over a decade now. I have one at home and one at work. I use it for gaming and have never had an issue with it....until a few weeks ago. I bought Quake 3 Arena off steam to get my rocket fix in, but I ended up getting a refund for it because the controls were totally off. If I was moving, I could not use left click to shoot anything, but I could move and shoot if I initiated the shoot first and held while moving. My guess was that the game patched something wrong, but I couldn't find anything on it so I gave up and refunded.
Ended up mostly ignoring that, until I bought Gauntlet Slayer Ed on GOG and I immediately noticed that I was having the exact same issue. Now I'm getting suspicious, so I unplug my mouse from my USB hub and plug it directly into the PC - issue still persisted. I was testing some things in-game and noticed that my other mouse buttons did not have this issue; I could right click and do the right click shot fine while moving and strafing, I bound an alt skill to another mouse button and that worked fine, I bound another mouse button to be left click and that was still broken.
So I uninstalled my mouse driver, and then the game worked just fine while left clicking and moving. Win10 has no native settings for remapping mouse keys, so I ended up installing X-Mouse Button Control, which is not a driver but it captures Windows mouse hooks and remaps them. Rebooting the game and it works totally fine using this; however, my mouse is 6-button and there is no way for the program to support more than 5 mouse buttons.
Anyways, this is annoying because I have never had an issue with either these mice or their drivers at home or at work. The only new thing at home that coincided with this starting was getting a new laptop at home that uses Windows 10. Both my work desktop and work laptop also have Windows 10, but they do not have this issue with games (and I've played multiple games on both of those machines). I'll report this issue to the mouse vendor, and maybe to MS as well because the same Mouse + Driver works fine on Win10 Enterprise, but not Win10 Home.
You're describing an issue that I thought only existed with trackpads. I've had to fix the same thing multiple times before, but have never had the problem using a mouse.
Are you playing on a laptop that has a trackpad? If so, try looking through your control panel settings. You need to find something called palm sensitivity in the trackpad settings, turn it off. I don't see how this would relate to the mouse, but with all the trouble I've gone through in the past with shit like this, it wouldn't surprise me.