I know air in a circuit will cause the gauge to fluctuate. If the circuit was left open for an extended period of time some moisture can accumulate, and the drier will need replacing. Check the HVAC compressor bolts, pulleys and belt tensioner. Over 20 yrs of working with Airco and never heard fog horn.
Also have you looked at the high side, there is a pressure cutoff on the discharge side of compressor lines. this shuts compressor off in over pressure condition.

 

The system was open about 4-5 hours while replacing the lines. Pulled vacuum for 20 min to -30psi.
Had to pull the front clip, radiator, fans etc to get the line in running under the engine.

The compressor isnt shutting off when it does this. Just the watching the gauge, it rapidly fluctuates.

Fog horn….its a deep groaning, honking, vibrating hose coming from the firewall line connection/expansion valve area. And the line gets hot. Google search didn’t pull much either about the noise either than on a few Fords.

 

Yeah, nothing remotely describing this. Which line or lines getting hot, the "small", liquid line feeds coil from condenser and of course the large one is suction to compressor. If suction getting hot at firewall most likely TXV is sticking, it appears to be mechanical valve not electronic. Before throwing parts at it, I would drop the charge blow out the suction/liquid line. unbolt the TX, blow through the coil in both directions. Gently blow out TXV. Reassemble, charge, and test. Not impossible but very unlikely you TXV failed after swapping hoses.
I feel you pain, mine crapped out the first week of hot weather, turned out to be clamp that held suction line to close to steering rack. Rubbed a hole in a 3 month old hose.

 

Yes the line that should be cold was hot.
As for the hose failure, yup down by the rack

 

This is at 80 degrees and it’s behaving itself. Think you might right with the sticking expansion valve

 

The fog horn is uncharted territory, diagnostics in the service manual for TXV is replacement instruction. Besides it's on our dime. Rockauto has several choices.

 

I heard a similar sound when my ford was venting Freon from a pop off valve somewhere in the system