I finally figured out how to get a LaTeX table of contents to look the way I want, with no dot leaders and with the page numbers just to the right of the section titles.

\renewcommand*\l@part[2]{%
  \ifnum \c@tocdepth >-2\relax
    \addpenalty{-\@highpenalty}%
    \addvspace{2.25em \@plus\p@}%
    \setlength\@tempdima{3em}%
    \begingroup
      \parindent \z@ \rightskip \@pnumwidth
      \parfillskip -\@pnumwidth
      {\leavevmode
       \large \bfseries #1~~\hb@xt@\@pnumwidth{\hss #2}\hfill}\par
       \nobreak
         \global\@nobreaktrue
         \everypar{\global\@nobreakfalse\everypar{}}%
    \endgroup
  \fi}
\renewcommand*\l@chapter[2]{%
  \ifnum \c@tocdepth >\m@ne
    \addpenalty{-\@highpenalty}%
    \vskip 1.0em \@plus\p@
    \setlength\@tempdima{1.5em}%
    \begingroup
      \parindent \z@ \rightskip \@pnumwidth
      \parfillskip -\@pnumwidth
      \leavevmode \bfseries
      \advance\leftskip\@tempdima
      \hskip -\leftskip
      \mbox{#1}\nobreak\nobreak\hb@xt@\@pnumwidth{\hss #2}\par
      \penalty\@highpenalty
    \endgroup
  \fi}
\renewcommand*\l@section{\@tocline{1}{1.5em}{2.3em}}
\renewcommand*\l@subsection{\@tocline{2}{3.8em}{3.2em}}
\renewcommand*\l@subsubsection{\@tocline{3}{7.0em}{4.1em}}
\renewcommand*\l@paragraph{\@tocline{4}{10em}{5em}}
\renewcommand*\l@subparagraph{\@tocline{5}{12em}{6em}}

 \setcounter{tocdepth}{1} % Only show sections and higher
\def\@tocline#1#2#3#4#5{%
   \ifnum #1>\c@tocdepth \else
     \vskip \z@ \@plus.2\p@
     {\leftskip #2\relax \rightskip \@tocrmarg \parfillskip -\rightskip
      \parindent #2\relax\@afterindenttrue
      \interlinepenalty\@M
      \leavevmode
      \@tempdima #3\relax
      \advance\leftskip \@tempdima \null\nobreak\hskip -\leftskip
      \mbox{#4}~~#5\hfill     \hfill\par}%
  \fi}
\makeatother