So, it’s true that the sun’s gravity is always pulling the Earth toward it. However, the way that orbits work is that the planet– in this case, Earth– is constantly ‘falling’ around the Sun. As an analogy: imagine you throw a ball on the surface of Earth. Then you throw it a little faster– it’ll travel a little farther before it hits the ground. Then you throw it even faster… as the ball moves faster and faster, it’ll get farther and farther before it drops. Eventually, as it falls, it will follow the curve of the Earth; if you throw it fast enough, it’ll just keep falling around and around the Earth. That’s all that an orbit is!

So, for the short answer to your question: the Sun’s gravity IS pulling on Earth, but only enough to keep it in orbit around it. There are other factors that affect how fast something orbits though, like larger nearby planets (like you mentioned). If something speeds up or slows down as it’s falling around the Sun, it’ll get farther or closer to it before it ‘falls’ back in its orbit.

Now, to my knowledge, Earth actually isn’t moving away from the Sun. However, the Moon is moving away from Earth! This has to do with tides and something called tidal friction. Another lengthy explanation… since the Earth is close enough to the moon, the force of gravity on one side of the Earth is much stronger near the moon and much weaker far from the moon– the difference in gravity is enough to stretch Earth out (the water of the oceans moves around to create these bulges, which is what causes high tides!). However, Earth spins faster than the Moon orbits (24 hours versus 30 days). Since those bulges take a while to smooth back out, it means that those bulges get ahead of where the Moon’s gravity ‘wants’ them to be. This creates these kind of gravitational handles that the Moon’s gravity tugs at, rotating it back in its spin– this is (very slowly) slowing down Earth’s spin, which is why Earth used to spin in 11 hours and now spins in 24 hours. However, when the Moon tugs Earth back, Earth tugs the Moon forward in its orbit so that it’s moving a little faster, which allows it to orbit farther from Earth. This is why the Moon is slowly moving away from us!