The Earth wouldn’t stop spinning entirely. Instead, it would eventually become tidally locked with the sun much like our moon is with Earth. Tidal forces very gradually reduce the rotational speed of an orbiting body until one side perpetually faces the larger body. So one half of the Earth would always be day and the other half always night. In this state, it rotates one time per year.
As for falling into the sun, that would take even longer as there is very little resistance to the orbital speed of a planet.
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