Q.Winter gardening
Should you still fertilize the winter crop like you do a summer crop?

My best advice is to know your soil and then know the needs of the plants you are growing.
A soil test is easy and will help you avoid over fertilizing.
Check the nutrient needs of the plants you have in your garden.
Seed packets have most nutrient information on the vegetables. Also you can search on the home page of Gardening Know How.
I also advise to go lightly on fertilizer when you do need it.
Here are some links to help you.
https://www.gardeningknowhow.com/edible/vegetables/vgen/the-southern-fall-vegetable-garden.htm
https://extension.colostate.edu/mg/gardennotes/711.html
https://www.gardeningknowhow.com/
https://www.gardeningknowhow.com/edible/vegetables/vgen/fertilizer-options-for-your-vegetable-garden.htm