'Heave a sigh of relief' does not mean 'escape from an unpleasant situation'. 'Heave a sigh of (whatever)' simply means 'to feel and perhaps express a feeling of whatever'.
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Ok, let's look deeper at the word resignation. Resignation is like a type of acceptance. For example. you chose to press on to achieve or accomplish something. Then, maybe you can use "Heaving a sigh of resignation, {so and so} went to do {something}.