Todo notes cropped (potentially with fullpage package)

The value to change is textwidth, and the manual points out that you need to adjust \marginparwidth, so

\usepackage[textwidth=0.7in]{todonotes}

Equivalently,

\usepackage{todonotes}
\setlength\marginparwidth{0.7in}

should do the job.

Do not use package fullpage. Set the margin with geometry:

\documentclass[a4paper, fleqn, 12pt]{amsart}

\usepackage[margin=2cm,rmargin=5mm,includemp]{geometry}
\usepackage{mathtools}
\usepackage{amsthm, todonotes, color}
\usepackage{showframe}

\begin{document}

    Lorem ipsum dolor sit amet,\todo{Lorem ipsum dolor sit amet, consectetur adipiscing 
    elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua.} consectetur 
    adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua.

\end{document}

The same setting as fullpage:

 \usepackage[margin=1in,marginpar=0.75in]{geometry}

or with fullpage

 \usepackage{fullpage}
 \setlength\marginparwidth{0.75in}

This is actually not so much a problem of todonotes. Rather, even a simple \marginpar would get cropped, and todonotes happens to use \marginpar. To avoid the cropping, you can make \marginparwidth smaller. (IMHO fullpage should do that automatically.)

 \documentclass[a4paper, fleqn, 12pt]{amsart}
 \usepackage{fullpage}
 \usepackage{todonotes}
 \makeatletter
 \setlength\marginparwidth{\FP@margin}
 \addtolength\marginparwidth{-4mm}
 \makeatother
 \begin{document}
 Lorem ipsum dolor sit amet,\todo{Lorem ipsum dolor sit amet, consectetur
adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna
aliqua.} consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore
et dolore magna aliqua. \marginpar{xyz abc def gh }

 \end{document}