Filling in Parametric Curve without Using pngplots

I am trying to fill in the area of two parametric curves like this:

Here so far is my failed attempt:

I would ideally like to not use pngplots. I know that pngplots comes with the fillbetween library (e.g. use case here), but I am wondering if there is a way to do this without it. I have been skimming the Tikz documentation, but have not been able to find anything. If there isn’t a good way to do this without fillbetween, so be it.

Here is a MWE used to make the pictures:

documentclass[crop, tikz]{standalone}

definecolor{axiscol}{HTML}{bebebe} % light gray
definecolor{fillcol}{HTML}{ffbf00} % amber
definecolor{arrowcol}{HTML}{cc5500} % orange amber

begin{document}

begin{tikzpicture}[domain = -2:2, variable = y, smooth]

    draw[axiscol, latex-latex] (-3.5,0) -- (3.5,0) node[right, axiscol] {(x)};
    draw[axiscol, latex-latex] (0,-3.5) -- (0,3.5) node[above, axiscol] {(y)};

    draw[color = arrowcol, very thick] plot ({sin(y r) + 2}, y) plot ({exp(-0.5*y) - 1.65}, y);
    fill[color = fillcol] plot ({sin(y r) + 2}, y) plot ({exp(-0.5*y) - 1.65}, y);

end{tikzpicture}

end{document}

documentclass[crop, tikz]{standalone}

definecolor{axiscol}{HTML}{bebebe} % light gray
definecolor{fillcol}{HTML}{ffbf00} % amber
definecolor{arrowcol}{HTML}{cc5500} % orange amber

begin{document}

begin{tikzpicture}[domain = -2:2, variable = y, smooth]

    draw[axiscol, latex-latex] (-3.5,0) -- (3.5,0) node[right, axiscol] {(x)};
    draw[axiscol, latex-latex] (0,-3.5) -- (0,3.5) node[above, axiscol] {(y)};

    path[fill=fillcol] 
        plot [smooth,samples=100,domain=-2:2] ({sin(y r) + 2}, y)  -- 
        plot [smooth,samples=100,domain=2:-2] ({exp(-0.5*y) - 1.65}, y); 
    draw[color = arrowcol, very thick] 
        plot ({sin(y r) + 2}, y) 
        plot ({exp(-0.5*y) - 1.65}, y);  
end{tikzpicture}
end{document}