I don't think you do, to an extent. I am just at the start of counselling to help me deal with the period my son was very ill (partially also heart, operated at 6 weeks old) and I've just been diagnosed with trauma-related anxiety.
I found I was in survival mode whilst everything was going on and when it was done, I was desperate for everything to be 'normal'. I now realise normal isn't always that realistic. My son has some developmental delays, nothing major, but enough for 'slumbering insecurity', as my therapist called it. It was an appointment with the GP for a referral to a paediatrician that broke me a few months ago, combined with some work issues I was having at the time.
You will find your normal, and the worry does get less and you'll find yourself at the GP's office less and less, but this is part of you and your family. Keep an eye on yourself. You may not need help now, but you might do later. Ask for it.
Also, the check ups will help. When you get good news, that the heart is fine (in our case if was a fixable thing with just one op) and it will lessen the worry.
But my main advice is to talk about it. Keep an eye on yourself and make sure you get the help you need whenever you feel you need it.