Sounds like you have style conversion issues (MS Word popularized style-based word processing, and, while powerful, it has its negative side effects). I used to work HelpDesk at a law firm and trust me, this problem happens an awful lot.
This might not be what you want to hear (as it will be a bit of a pain), and I doubt these are the only ways to fix it, but for what's it's worth, these are 2 ways I know(1. is less drastic, but might not work. 2. will fix it for sure but will take you a while).
1. Go into the styles menu and set the Header 1 style and the Normal Style as what you want them to be. This includes Font, Spacing and Indents. (I'm going to assume that all you have is chapter headings and body text). Then go through your document and make sure that your headers are associated with the Header 1 style, and your Body Text is associated with the Normal style. Then try converting to .rtf. (Note: if what you just read sounds like gibberish and you do not know where the style menu is, I would say - bite the bullet and go for option 2).
2.Take the entire document and remove all formatting (painful, I know - and save a backup copy before you do anything). In MS Word there's an eraser button in the formatting tab that does this. However, in all word processors, if you 'cut' all the text, then Paste Special and pick Unformatted Text Only, it'll do the same thing. Save the new document (with the text only) as an .rtf file. Then, inside this new .rtf file, go back and reformat your document as per the publisher's requirements.
*Things to watch out for - tabs at the front of paragraphs and/or lines between paragraphs. From the 'problem documents' I have seen, what people tend to do is be inconsistent when defining paragraphs. For example, some paragraphs are indented using the automatic 1st line 0.5 in indent (which is part of the style, usually), and some paragraphs are indented using a physical tab in front of the paragraph. When reformatting as I described above, you will need to go back and fix this (i.e. reformat your body text to have a 1st line 0.5 in indent, then go back and delete some of the physical tabs that you might have accidentally put in.)
**If everything you just read sounds like gibberish, ask around for someone good with MS Word/Open Office and ask them to help you fix it. Honestly, that's what the secretaries at the firm I worked for did. It would take a mid-level MS Word user 15min to fix the problem you just described, but it might take someone less familiar with the program 2-3 hours and cause much frustration.
***For future manuscripts: To avoid this problem, I work exclusively on .rtf files. I have discovered that almost all publishers prefer .rtf files, and they don't want any formatting beyond double spaced, 1st line indent, with italics for emphasis. Most don't even want chapters centered and/or in bold. Both MS Word and Open Office give you the option of saving as .rtf from the get-go (they default saves to .docx etc., but you can always make it save to .rtf instead. This way, if there are style formatting issues, you will see it right away instead of after the fact.
****That said, Track Changes and .rtf files don't play nice and will result in sporadic crashing
Hope this helps!