Everything should be fine since she stopped coughing and doesn't seem distressed. The fact that she was eating and drinking right away makes me happy. By all means, get reassurance from your vet about any concerns you have.
It's strange how different each bird can be about taking oral medicine. Giving her tiny amounts slowly (as you are) is about the best way to keep her from getting it into her airways. She can "handle" the liquid if there's just a bit of it at a time to avoid breathing in. She won't be overwhelmed by the volume.
The most important thing is not to restrict her chest movement so she can breathe. Your husband's skill as a wildlife guy should be taking that into account. Your vet and the videos should have addressed that as well. Holding securely usually gets them to stop squirming. They know they can't get loose no matter what they do. I try to keep the wings tucked flat in the normal "resting" position and have their feet gripping the towel ("tucked", not flailing about). I (and my vet) always hold the bird until they calm down a bit before dosing (so they are not trying too hard to breathe). My best trick is to let the bird bite something (a corner of the towel, a small stick) after they have some medicine in their mouth (if necessary or things are taking way too long). It seems to get them to forget about keeping the medicine out of their throat and normalizes their reactions and swallowing. Maybe they are just happy to be BITING something
. Moving that bottom mandible seems to help getting the medicine down. I have one guy I must let bite something (the towel) just to get his beak open enough to trickle in liquid. These things don't seem to be your problem, though. (Yet.
Some of them get pretty crafty. I had a macaw that would plug the tip of the syringe with his tongue! Little bugger!). Paying close attention to YOUR bird will give you the best clues on how to proceed. I can really relate to "moving the bird around" to get the medicine down. This usually involves tipping them back to get the liquid to trickle into their throat from under their tongue.
Praising them and treating them well after they have taken their dose is always a good idea!
P.S. Chucki's "head lock" is essential, concentrating on keeping the "skull" part of the head steady. That post above is ALL good!