Air Carrier Captain reported gate departure without the required dangerous goods form due to a communication breakdown between flight crew and Dispatcher. The high workload of multiple issues and time pressure at departure time caused the communication issue.

Narrative

The difference between this flight and a LOFT is that in a LOFT; we know when it is going to end because there's an end to the SIM availability. In retrospect; this was a case study in additive conditions. Planned at Allowable Takeoff Gross minus 100# with a short flight and an alternate. Attempting to accommodate a Jumpseater; HAZMAT; multiple releases; I traded into the leg to help a fellow Pilot. Unusual day; because of incoming weather. The last two days of my trip were pulled; due to cancellations. Requirement to burn fuel; prior to push; ZFW issues; OTP pressures; I was notified; in passing; that there was HAZMAT on the flight; during a conversation with the Operations Agent and Dispatcher about Allowable Takeoff Gross and ZFW issues. The amendment to add the HAZMAT was sent via ACARS. After departure; I realized that I had not received a Dangerous Goods Form. I seem to remember that the HAZMAT was pulled; but cannot say for certain. If that was the case; then I should have amended the Release to remove the note for HAZMAT. If the HAZMAT was still on; I failed to ensure the Dangerous Goods Form was delivered to the cockpit and in the logbook.