Although in the game, escorts will stick with you even after running out of depth charges, by 1944 this was a legitimate ASW tactic for Allied navies. As long as the attacker can maintain contact, they can hold the U-Boat down until battery exhaustion or CO2 build up force the boat to the surface where it can be destroyed by gunfire. One of these attacks lasted more than 36-hours and boredom was unlikely to have been a problem for the crew of the target boat.
Escaping detection in shallow waters against a single destroyer is difficult but not impossible. Also, the removal of the depth charge threat allows for a counter attack. In October 1944 you have access to acoustic-homing torpedoes, either T-IV Falkes or T-V Zaunkoenigs and if you leave for patrol with less than four, or all that are available, you're doing late-war patrols wrong. Usually I have two acoustic torpedoes aft and one with a reload forward. Don't waste them on merchants as they can save your butt in situations like you found yourself. This leaves you with just eight torpedoes for anti-shipping operations but can save your boat.
In your case you an acoustic shot into the stern of the DD after it passed overhead and from close range might have been effective, even using the inferior Falke. You don't even need to sink the attacker; crippling it allows you to get away successfully.
-C
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