Rather than use \large
and friends, you want to set the font size and baseline skip "by hand"
\fontsize{1cm}{1.36cm}\selectfont
The first size is the font size, you will just need to choose that by eye, depending on which font you are using. The second is baselineskip.
You also then want to set \parskip
to a multiple of \baselineskip
so it does not push you off grid at a new paragraph. That is, in your preamble go something like
\raggedbottom
\newlength\myline
\setlength{\myline}{1.36cm}
\setlength{\parskip}{0\myline}% this puts no vertical space between paragraphs
%\setlength{\parskip}{1\myline}% uncomment this if you want exactly one blank line
If you just have text and paragraph breaks, that should be enough. If you also have (say) itemized lists you will need to similarly set spaces such as \itemsep
to be a multiple of \myline
so that the itemized points still end up between your lines.