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It sounds like you want to keep the doors, so that you can open them in good weather.
Another option to replacing them is to winterize with insulation that can be added in colder months and removed in warmer months. I've seen products on Amazon that are basically foam board with a backing that stick in panels.
Of course, this means that you can't take it down on a warm day and you have to take it down in spring and repeat.
Regardless of what you do to the door (new door or winterize), you will need to figure out a way to prevent transfer of heat around the door. These must have large gaps at the top, bottom and sides. That means a threshold barrier and likely tape around the rest of the edges (inside and out).
Just another option for your budget.
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