Okay, bear with me a sec while I write down my thought process...
So, your DVI cable broke and then you switched to HDMI.
The main difference between HDMI and DVI is that HDMI carries sound, whereas DVI does not.
Your sound works fine while using the monitors speakers (which is using the HDMI signal).
Your sound doesn't work properly while using the HDMI with an external speaker (head phones).
Your motherboard has a sound device on it, which you connect your speakers to, but you also tried connecting it to your TV to no avail.
Your GPU has an onboard sound device, which enables the sound to be carried over the HDMI cable (of which works fine).
At first it sounds like there's a device conflict, but it could just be some settings, for example,
Voice Cancellation (pic), so check that in your sound settings.
Also if you have a sound program like Realtek HD Audio Manager, then i'd check those settings out, to see if there are sound effects applied to it.
You also want to make sure that your sound settings are either on 24bits, 48,000 Hz or 16bits, 44,1000 Hz - if it's already on one, try the other one.
If none of that works out, then I recommend trying to update your GPU drivers and your sound drivers.
If it doesn't sort itself out after updating your drivers, try disabling the AMD High Definition Audio Device in
Device Manager (pic).