EMB-175 Captain reported failing to get a new release or request priority handling while diverting for a hydraulic system problem.

Narrative

While holding; preparing for a diversion to ZZZ from our original planned route (ZZZ1-ZZZ2) due reaching our calculated bingo fuel number; we received an advisory EICAS message for Hydraulic 3 low quantity. Once we received this message as a crew we began acting with a heightened sense of urgency. While not requesting priority handling as the system was still operating normally with pressure in the desired range we decided we wanted to be on the ground sooner rather than later as our Hydraulic 3 quantity was showing as 18% and dropping. We communicated our situation and intentions with our dispatcher via ACARS and came up with a plan. ZZZ was determined to be the most suitable alternate in the situation primarily due to its close proximity. While descending into ZZZ; setting up for the approach; and managing our hydraulic issues; we overlooked the fact that although we had positive communication with Dispatch and agreed upon a plan; we never received our new release with numbers for ZZZ on it. We ultimately landed at ZZZ without this new release; and without requesting priority handling; hence an error in operational control.Cause: Primary cause would be tied to task saturation in managing both our descent and approach into ZZZ; as well as our hydraulic situation. This would be the cause for why the need to receive a new release was overlooked.Suggestions: Slow down; ensure we have all proper information for what the situation presents. Potentially easier link for communication with dispatch as we had multiple difficulties with communication due to datalink reception being spotty.