I am having so much trouble with Windows.
I was wondering if I could get a suggestion as to which version of Linux would work best with my mobo? Its a AMD dedicated board.
A Gigabyte; GA-MA785GM-US2H (rev. 1.1). It has the two onboard chip sets for Video and Audio. A North Bridge: AMD 785G and South Bridge: AMD SB710. I also have the AMD Phenom II x6 1090T as my cpu and onboard memory of 2Gb x 4.

I was forced to totally reinstall Win7 after a major malfunction.
My biggest problem is that no matter what I do, I can't get the HDMI output to send both sound and video to my TV or my computer monitor.
I get no sound at all. Its driving me crazy that I can't get windows 7 ultimate x64 sp1 to let me install HDMI drivers for the onboard sound. Now Microsoft doesn't support Win 7 ultimate x64 w/Sp1, I am over it.

So I was wondering about switching to a Linux OS. What are your recommendations with this board and cpu, as there is onboard video and audio?

Mint,MX Lubuntu,Xubuntu. Try any of those to give you a semi-Win 7 feel and they'll run on whatever you have.

This desktop has an AMD A6-9500 on a Gigabyte A320M-S2H-CF board. It runs PCLinuxOS, Xfce version, very happily. I don't use the monitor for sound, but a quick check shows that I have an HDMI driver that was automatically loaded into the kernel. With Linux, such drivers are provided by the Linux Foundation as plug-in kernel modules and you don't have to deal with them yourself.

As far as my opinions go: https://wiki.debian.org/DebianExperimental
...I know it's hard to make your own at this stage so thank them in a few years!

More links:

The ones Bonzoo said are very good. Especially Linux Mint. I've always found it easy and it works very well out of the box. I dumped Windows 10 in November 2015 after experimenting with Mint in a dual boot. Since you have 8 GB RAM you should have no problem with the Cinnamon desktop. I run it with 4 GB. If you're going to try a desktop like GNOME, which Mint doesn't carry, you may find that one a bit counter-intuitive. Desktops (DE's) like MATE and Xfce are like Cinnamon, but with fewer features.

If you don't have a program like Rufus or Etcher, I would start by downloading that in order to flash the iso file you will download of the Linux distro(s) of your choice. You can then try them out in the Live Environment before you decide to install it.

If it works well Live, then it should be just as good if not better after a full install.

If you decide to dual boot, before completely dumping Windows, you should identify if your BIOS is the old MBR or the newer UEFI. The newer UEFI there are some additional steps to make a dual boot work, like disabling Fast Boot. There's another setting I had to change as well. For comparison purposes, my Compac machine from 2008 uses MBR and my Lenovo from 2012 uses UEFI.