How should I upgrade my equipment?

The easiest way is if you can find the DTV hard drive kit that turns the H44 into an HR44.

Call DTV it is possible they may still have one.  Call or log into your account to order a free upgrade but most likely it will be to an HS17 Genie2 with a new 2 yr commitment.

With the Genie Lite (H44) you would need to order the DVR upgrade kit. This is a proprietary external hard drive for DirecTV that makes it act like an HR44. If they have any of those available as the Lite was a "good" idea that didn't last long (caused a lot of confusion, especially from 3rd party dealers that mislead customers about function and cost).

Most likely the option would be an upgrade to the Genie-2 (HS17). This would not go on a TV itself, so a Mini Genie would go where the H44 is now (possibly a C61). You would have 7 tuners to share for watching and/or recording. It also has 2TB of recording space.

The upgrade would come with a 24 month agreement. Bill should add $10 DVR and $3 Whole Home DVR, or just $15 Advanced Receiver services, depending if you're on the old or new billing model. You would have to call DirecTV to verify if you qualify for a free upgrade. Expect to schedule an appointment.

The minis require Whole Home so there is no additional fee.

Genie Lite without the DVR upgrade kit doesn't charge the DVR service at the very least. That was the issue where 3rd parties would sell new service claiming DVR without the fee, not informing the customer that DVR was optional.