There are no "guarantees" and there is no way for you, nor anyone else, to "know for sure" whether a newer version will run better on "your" system until you install it on "your" system.
No one here knows what all apps & browser extensions you have installed, how many apps are running, what web sites (if any specific ones) are causing your issues, if your problem may be network related vs flash related, how much free space you have on the drive, how much RAM you have in the machine (or how much is free at the time of your problems), nor at this point even which Mac you have.
You're running a 5 yr old OS, 3 versions behind the current one.
You're running a version of flash that is several versions behind - you admit it doesn't work and locks up your system - sounds like you have nothing to lose by installing the most current version Adobe offers for 10.6.8.
In any case, you do what you should be doing prior to installing any new software on your computer - you make a current backup - then install the new software - test it - don't like it, don't want it - restore your machine from the backup and you're back to the old non-working version of Flash.