I suspect that what is bothering you is the lines from a table that are showing up.
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These lines appear in a pdf version produced by Word or by Adobe Acrobat. They do not appear in print preview.

I do not really know why one table has this happen and another does not. The table that is showing up appears to be floating behind text. I can't find it in the header/footer layer. However, there seems to be something in your header that is really distorting things. I can't find it but can see that something is weird.
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There is something making the header area extremely large. What is anomalous is that this header area is not pushing your text down in the body. The parts of your document's body which overlap are in floating objects such as text boxes and floating tables.

In the body, even deleting the table and text that is on top of it does not fix this.
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Changing the setting to "different first page" in the header does not solve the problem, in that the first-page header then has the spacing that your header had.

Was your resume originally produced in some program other than Word?

I tried copying everything but the final paragraph mark (which holds headers and footers) into a new document. Same problem.

As written, your resume is extremely difficult to edit. It is a real mishmash. Headings are in text boxes rather than in Headings in the body. The spacebar was used for spacing rather than using tabs or the table structure. In one spot (Projects) a textbox is used rather than a table. To me it is a minor miracle that this looks as good as it does.
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I would recommend sitting down with a friend who really understands how Word works and starting from scratch. Some tips.

1. Display non-printing formatting characters. Showing non-printing formatting marks in Microsoft Word
2. Use Tables for arranging text horizontally. These can be Floating Tables.
3. Do not put headings in text boxes. Instead use a heading style with a bottom border if needed.