Metadata not saving to original file

"I just want to make sure my copyright info is in the metadata before I send it to other people, so would it be best exporting files I need to send from lightroom to ensure this."

Exporitng to DNG, TIFF, or JPEG file format will contain the copyright information you've entered in the LrC Metdata panel. As mentioned raw files use XMP container files for all metadata and edits added inside LrC. When sharing files it's best to use TIFF or JPEG file format unless you want the recepient to do further editing of the files in LrC. In that case DNG is the best file format as it will contain all of your editing information.

Lightroom Classic by default does not make changes to the original imported image files, in this respective it is nondestructive.

Lightroom stores all the edits and other functions you perform in the Catalog file,

When you're working in Lightroom as far as Lightroom is concerned the info is in the Catalog file and Lightroom is reading and writing from the Catalog file. There is no option available to alter this process.

When you import image files Lightroom will read info from the file header or an XMP sidecar if available and that info will be recorded in the Catalog file for future use.

There is an option available to “Automatically write info to xmp” (note this is write-only). The main purpose of this function is to allow other Adobe applications like Bridge, Adobe Camera Raw and Photoshop have access to the main functions you perform in Lightroom. Info that unique to Lightroom like Collections, Virtual copies, Published images etc are not written to the file.

There is also an option available to manually read and write info to the files in the Library Module under the Metatadata menu.

The likes of Bridge, Camera Raw, Photoshop cannot read or write to the Lightroom Catalog file.

Just some general info.

Automatically write changes into XMP. 

Sometimes "Automatically" is not always 'instant'. LrC may wait for a 'quiet' time to write the metadata. It may even pause writing metadata if LrC is shutdown and resume when re-started.

saving to original file....now a xmp file next to all my files

Proprietary raw files (NEF, CR3, ARW, etc) will always have a sidecar .XMP created by [Ctrl+S].

Raster files (JPG TIF) will have the metadata saved within the original file (no sidecar file. Original is modified.).

by then exporting the images back to the folder but then it creates yet another file

Well, that is to be expected! You are creating a new derivative image file by 'Export'.

Explain exactly what you are trying to achieve by writing metadata to files. And what file format are you images?

"there is now a xmp file next to all my files as well,"

For non-raw files (e.g. JPEGs), LR writes metadata into the file itself. But for raw files, LR writes .xmp "sidecars" that contain the updated metadata.

Which types of files are your working with?  How are you determining that the metadata is saved?