I do something very similar. I’m a bit of a shut-in, so verbal communication is rare, particularly extended communication. If I can stay silent, I will do so. Had this happen one time, and it took me about 3 weeks to get over completely, but I took it way slow. Still go through bouts of complacency and forget that I need to speak.
Getting to the helpful part- I have to back up what fluffybunywhiskers suggested. It’s similar what I do now, except with a bit of variation. I find humming works quite well as a warm up before actually talking, or even in down time when there’s nothing to be said, and after that I tend to talk during a game, book, etc. Just outing my thought process so I don’t have to really think about talking, I can just do it when I know I need to.
I can recommend: picking up humming often as a habit, speaking words to songs on the radio/phone/Ipod/etc (I feel it’s safe to assume you don’t sing often, so this would be a better starting point than that), narrating games, following trains of thought aloud, and reading aloud.
Of course, it can take some time to remember to actually talk often enough to keep your voice in working order. It’ll also sound odd talking to yourself, but it helps tons.