Is it true that wearing glasses will actually worsen your vision when you start wearing them?

Not at all. This is an old wives' tale. Your eyes' prescription is a result of their shape and normal changes that occur over time. This built-in shape is not affected by outside sources (except for injury/infection). Glasses correct the prescription that the eye has; they do not alter the eye's shape. As your prescription naturally changes, you will need a stronger prescription, and not the other way around.
No. My personal example - I've worn eyeglasses for 13 years without significant change in my vision or prescription. If a worsening change occurs, it may be due to the worsening condition or natural aging that is age related.