Earth is estimated to host life for at least another billion years, if no disasters that could destroy Earth happen.

The moon is indeed getting away from Earth, at the rate of 3.8 cm/year from Earth. But no worries, it won't impact us in the near future. Assuming the rate stays the same (3.8 cm per year), the closest planet that the moon will probably hit is Mars, which is 5.46*10⁷ kilometers (or 5 460 000 000 000 cm) away from Earth. So based on the simplest assumption, it would take 1437 billion years before the moon hits Mars. This is definitely not the concern for us, because our sun will only last for another 5 billion years before it runs out of fuel.

Just keep in mind, the moon and other planets are actually quite far away from us. The moon is much closer to us, with an average distance of 384 402 kilometers from us. Even if we assume it is moving toward us at the same rate of 3.8 cm per year, it would still take 10 billion years before it hits us. Of course, the assumption of the same rate of 3.8 cm per year is not true forever because the magnitude of gravity depends on the distance of two subjects, which would otherwise influence the rate for Moon to move.

There are other concerns that you should consider as the end of the world. The first one is the war between us. We already have the ability to destroy our Earth, or at least make it uninhabitable for life using nuclear weapons alone. Some other disasters could also end the world such as the slower or abrupt climate change (it could take thousands or millions of years, but still it is much shorter compared to the above estimation of billion years), the unexpected asteroid hitting Earth (could happen anytime in principle).

Wikipedia has an article about your question:
future of Earth .

The article says in 1 billion yrs, about 27% of the ocean will go under the surface of the earth.

It also says the Moon and Earth might collide in about 65 billion yrs, if they haven't both been destroyed by the sun by that time.

The moon is moving away from the earth as the earth slows down its rotation. When the earth's rotation equals the orbit of the moon, the moon will stop moving away. Unless some other planet disrupts the earth-moon system, the moon will not hit another planet.