The idea is to set the original indent (the indent to be used for the (sub)sections in the document) in a renewed \tableofcontents, and then override these values as soon as the \appendix command is encountered in the document.
Here's one possible solution, assuming that you are not using and will not use \paragraph. Copy the code between the %%%%%%%%%%%%%%% lines in the preamble of your document:
\documentclass{article}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcounter{mysection}
\renewcommand\themysection{\Alph{mysection}}
% Writes the entry in the ToC with the required indentation
\newcommand\ToToC[1]{%
\addcontentsline{toc}{section} {\hspace*{4em}\appendixname~\themysection\hspace*{1em}#1}
}
\makeatletter
\newcommand\mysection{\refstepcounter{mysection}%
\@startsection{paragraph}{4}{\z@}%
{-3.5ex \@plus -1ex \@minus -.2ex}%
{2.3ex \@plus.2ex}%
{\normalfont\Large\bfseries}}
\renewcommand*\l@section{\@dottedtocline{1}{0em}{4em}}%
\renewcommand*\l@subsection{\@dottedtocline{2}{0em}{4em}}%
\renewcommand*\l@subsubsection{\@dottedtocline{3}{0em}{4em}}%
\newcommand*\l@mysection{\@dottedtocline{4}{1em}{1em}}%
\renewcommand\appendix{\par
\setcounter{section}{0}%
\setcounter{subsection}{0}%
\renewcommand\theparagraph{\Alph{mysection}}
\renewcommand\themysection{\@Alph\c@paragraph}}
\makeatother
\setcounter{secnumdepth}{4}
\setcounter{tocdepth}{3}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
\tableofcontents
\section{Test section one}
\subsection{Test subsection one one}
\section{Test section two}
\subsection{Test subsection one two}
\subsubsection{Test subsubsection one two one}
\appendix
\mysection{Test appendix one}
\ToToC{Test appendix one}
\mysection{Test appendix two}
\ToToC{Test appendix two}
\end{document}
After the \appendix command, use the \mysection command instead of the standard \section command and immediately after, use the \ToToC command to include the entry in the Table of Contents. This solution will not work well if the appendices have subsections, though.