You may have astigmatism that is insufficiently corrected by your glasses. Astigmatic eyes have multiple focus points that create double images. The high contrast of the moon against a dark sky makes the fainter "ghost" image more visible. If this is the case, you may be able to replicate the effect with a high contrast image, such as black text on a white page under bright light.

Vision also tends to worsen with age (there are some exceptions that temporarily improve). Prescriptions for astigmatism have two diopters (sphere, cylinder) plus an angle (axis). Some contact lenses for astigmatism are weighted to prevent rotation from affecting the axis. (Some contacts use a different correction method that allows them to ignore the axis.) Also, if you tilt your head during the eye exam, the axis will be off and the prescription won't be quite right. Whether you need new glasses depends on your tolerance for slightly imperfect vision.

If you use contacts with the "same" prescription as your glasses, and the problem occurs only with your glasses, the problem is likely something associated with your glasses, such as reflections, as Michael C suggests.

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Some medical conditions and substances (alcohol, THC, antihistamines, medications, etc) can create similar effects by causing the eye ciliary muscles to pull unevenly on the lens. If substance induced, the effect is likely to be temporary.

If you have other medical conditions that can affect vision (including diabetes and hypertension) you may need an eye exam regardless of the need for new glasses. If your vision rapidly worsens, with or without pain, it could be a medical emergency (such as glaucoma). Use your best judgment. Consult a physician.