See http://www.word.mvps.org/FAQs/TblsFldsFms/LinesInForms.htm 
and http://www.word.mvps.org/FAQs/Customization/FillinTheBlanks.htm .

Did you use fields from the forms toolbar? Are you using the tab key to move
from field to field?

Generally people put form fields in table cells or frames that are not
allowed to change size if they do not want material inserted at a field to
change the position of other elements.
http://www.word.mvps.org/FAQs/TblsFldsFms/LinesInForms.htm.

What you are talking about is what Word calls an "online form." For more
about online forms, follow the links at
http://addbalance.com/word/wordwebresources.htm#Forms or
http://word.mvps.org/FAQs/Customization/FillinTheBlanks.htm especially Dian
Chapman's series of articles.

Hope this helps,