Receiver upgrade recommendation.

AFAIK DTV is only offering a Genie and minis as an upgrade most likely an HS17 Genie2 which is a client only system.

To see if you are eligible log onto your account at Directtv.com  look under “order new equipment” to see if you are “eligible” for a free upgrade. 
Once you have that information you will need to call customer service @ 800 531-5000 They are understaffed so be prepared for a long wait. 

Forum users are not employees of AT&T or direct tv 

What are the models of your other boxes? (find it unlikely the entire account would be one room).

If those 4 are truly all you have...

Upgrade to Genie. This is a 5 tuner, 1TB, HDDDVR. Normally a free upgrade to your first Genie would include up to 3 Mini Genie Clients (the most that can be in use at one time). That would allow you to get the benefit on all 4 of your TVs.

Clients have no tuners of their own, each taking 1 from the Genie. So you would still have 1 tuner more overall than you have now. Recording option replaces the Autotune feature. Even if you don't replace all your non-DVRs, each TV can watch what you have recorded.

Since you should already have $10 HD service, this would add $10 DVR and $3 Whole Home DVR. And of course all upgrades come with a 24 month service agreement.

On Demand requires connecting to your home internet. The 2nd/3rd generation Genie (HR44/HR54) have a wireless option built-in that can use 2.4Ghz or 5Ghz. However, wired is preferred such as direct Ethernet, BB Deca that connects coax to router, or even Ethernet by way of a powerline adapter if needed.