Air carrier Captain reported descending below the final approach fix crossing altitude and receiving an advisory from ATC. The flight crew then returned to the correct altitude.

Narrative

While descending into ZZZZ ATC asked us to keep our speed up and gave us several short cuts on our final descent. We were cleared to 4000 feet and we were almost past ZZZZZ the initial approach fix for the RNAV XX. At this point we were cleared for the approach. We were passing 4000 feet and just under 250kts indicated. I was using level change to keep the autopilot from slowing us as we were asked to go fast before for traffic behind us. The final altitude on the approach is 3300 feet. As we were slowing I called gear down and for flaps 1;2;5 then 15 then manually set speed 180. As we turned final I thought I had pressed the VNAV button. As we descended below 2700 feet just outside of the final fix ATC announced we were below 3300 feet. I saw we were low just an instant before ATC and saw we were still in LVL CHG. I initiated a climb to 3300 feet and crossed ZZZZZ1 (FAF) at 3300 feet and initiated a VNAV descent to a landing.Cause - ATC asking us to go much faster than normal in the terminal area and several short-cuts compressing the time to configure for approach. I was also still becoming accustomed to the 737NG screen layout as they were recently changed on [two aircraft]. After finding we were below the final altitude began a corrective climb using VS mode. It is imperative to always double check the FMA before assuming the mode.