So anytime I have seen QOS implemented in a cheapo router, it slows down everything.
These types of devices don't have the power to do traffic shaping on a ton of devices.
I would take a look at the wifi settings and see what channel it is broadcasting on.
You may need to play with the channel to get to one that is less saturated, basically you want the one that your neighbors are not all using as well.
Use a free iphone app or android app to test signal strength while walking around your house and see what works best.
I would not buy a router from verizon, they are going to give you garbage. If you need to get a better router, check the reviews on newegg and buy one there.
You will need to set the verizon provided device to bridge mode and physically uplink a cable to the new router.
How big is your house also?