What can I do to fix my glitches? (Skyrim)

I have been playing Skyrim and the more I play the more glitchy my game is getting. I play on xbox 360. Note: I play 100% Vanilla with no DLC's

Examples:

• Horses start climbing trees
• Dragons flying inside buildings that are out of the rendered Skyrim world.
• My follower floats 1 meter off the ground like it keeps repeating the falling animation while flying.
• I sometimes spawn next to dragons that just instantly kill me when I spawn.
• I walk around and I see floating swords that are intangible to my character.

The most recent of examples was when I was shooting at a mammoth behind Fort Greymoor. I shot the mammoth once and it decides to just fly off high up in the sky.

This is very frustrating to me because it prevents me from finishing certain quests. I keep doing the dawnbreaker quest (guide meridias light through the temple) at the end of the quest when I take the dawnbreaker you meet her in the clouds and she is supposed to talk to you. On my game she never appears and then after 5 minutes of being in the clouds I fall down to Skyrim and die. I tried fast travelling but it wouldn't let me.

Is there a way to fix these glitches for the xbox 360? Or do I have to rebuy the game?

Thanks