Arrows in tikz Markov chain diagram overlap

bend left and bend right come with parameters, the bending angles. Adjusting them allows you to avoid the intersections. (BTW, I also removed packages that were not used. Note also that the arrows library got superseded by arrows.meta but I kept arrows for now.)

\documentclass[reqno]{amsart}
\usepackage{tikz}
\usetikzlibrary{automata}
\usetikzlibrary{positioning}  %                 ...positioning nodes
\usetikzlibrary{arrows}       %                 ...customizing arrows
\tikzset{node distance=4.5cm, % Minimum distance between two nodes. Change if necessary.
         every state/.style={ % Sets the properties for each state
           semithick,
           fill=gray!10},
         initial text={},     % No label on start arrow
         double distance=4pt, % Adjust appearance of accept states
         every edge/.style={  % Sets the properties for each transition
         draw,
           ->,>=stealth',     % Makes edges directed with bold arrowheads
           auto,
           semithick}}

\begin{document}

\begin{figure}[htb]
\centering
\begin{tikzpicture}
\node[state] (s1) {State 1};
\node[state, below right of=s1] (s2) {State 2};
\node[state, below left of=s1] (s3) {State 3};

\draw (s1) edge[loop above]  (s1);
\draw (s1) edge[bend left]  (s2);
\draw (s1) edge[bend right]  (s3);

\draw (s2) edge[bend left=12]  (s1);
\draw (s2) edge[loop right]  (s2);
\draw (s2) edge[bend right=12]  (s3);

\draw (s3) edge[bend right=12]  (s1);
\draw (s3) edge[bend right]  (s2);
\draw (s3) edge[loop left]  (s3);

\end{tikzpicture}
\end{figure}
\end{document}

you can reduce default value of bend angle. just add bend angle=15 to your tikzset (similarly @marmoth change it locally for two arrows bend).

off topic:

• for labeling of arrows is handy to use quotes library and than wrote it as for example ... (s1) edge["label",bend left] (s2).

• package hyperref had to be load last in preamble (except in rare cases)

  \documentclass[reqno]{amsart}
  \usepackage{amsmath, amssymb}
  
  \usepackage{pgfplots}         % it load tikz too
  \pgfplotsset{compat=1.16}
  \usetikzlibrary{automata,
                  arrows.meta,    %   ...customizing arrows
                  positioning,    %   ...positioning nodes
                  quotes}         % For edge labels
  \usepgfplotslibrary{fillbetween}
  \tikzset{node distance=4.5cm,   % Minimum distance between nodes. Change if necessary.
           every state/.style={   % Sets the properties for each state
                  semithick,
                  fill=gray!10},
           initial text={},       % No label on start arrow
           double distance=4pt,   % Adjust appearance of accept states
           every edge/.style={    % Sets the properties for each transition
                  draw,
                  semithick,
                  -Stealth,       % Makes edges directed with bold arrowheads
                  auto},
           bend angle=15          % Reduce default bend angle
           }
  
  \usepackage{hyperref}           % had to be last in preamble
  
  \begin{document}
      \begin{figure}[htb]
      \centering
  \begin{tikzpicture}[]
  \node[state] (s1) {State 1};
  \node[state, below right of=s1] (s2) {State 2};
  \node[state, below left of=s1] (s3) {State 3};
  
  \draw   (s1) edge[loop above]   (s1)
          (s1) edge[bend left]    (s2)
          (s1) edge[bend right]   (s3)
  %
          (s2) edge[bend left]    (s1)
          (s2) edge[loop right]   (s2)
          (s2) edge[bend right]   (s3)
  %
          (s3) edge[bend right]   (s1)
          (s3) edge[bend right]   (s2)
          (s3) edge[loop left]    (s3);
  \end{tikzpicture}
      \end{figure}
  \end{document}