It is likely that your resumes are .docx files. DOCX and ODF (the LO internal format) are quite different and can’t fully be converted to each other.You seem to hint that your resumes are not exclusively “text” but also contain “decorations” in the form of text boxes. This will cause major compatibility issues.
My best advice is to recreate your documents from scratch as .odt files, using native Writer features such as styles and [text] frames (if you really can’t design your document as “pure” text). But I warn you that frames are really, really (should be repeated seven times) difficult to tame.
Think deeply about your document architecture. Are “text boxes” necessary? Perhaps a table will give you better and simpler control on your elements. Table cells are complete sub-documents by themselves. Pay special attention to M$ Word workflow: it encourages direct formatting because of lack of adequate features/primitives. Writer is different from Word. Try to adopt the appropriate approach from the beginning, i.e. styling.
To spare you some pain, you can “import” your existing text by copying from your resume and pasting it as unformatted in the new document. Then apply styles to format it.
Before diving into the task, I highly recommend you read at least the Writer Guide and also the excellent Bruce Byfield’s Designing with LO, available from the same link after pressing the “More” button and scrolling down.
