While it is difficult for mosquitoes to actually bite our pet birds due to the feathers protecting their bodies, the foot and eye area is exposed and can be bitten by the annoying mosquitoes.
And yes, the buzzing around made by mosquitoes can very well trigger night frights on our pet birds.
In order to protect yourself and your pets, you can place mosquito nets on all windows of your house, the same goes to all doors leading to the outside.
You can also cover your budgie's cages to prevent the mosquitoes from getting inside and disturbing your budgies.
I too am very afflicted by the mosquito bites, unfortunately my blood type is their favourite and over the years I have become quite the mosquito catcher.
I don't use any repellents nor sprays, I mainly catch them with a fly swatter when they have landed or my bare hands when the mosquitoes are mid fly close to me. I have zero tolerance with these creatures and get deeply focused on their flying patterns to catch them while flying.