KOH in methanol will hydrolyze the glycine. -OH is a better nucleophile than -OEt, and once it is attached, it is deprotonated by the base and no longer undergoes substitution. I don't believe you can make esters with hydroxide bases in any solvent system.
Also, although the extraction would work with most simple carboxylic acids, I don't believe you would be able to extract your glycine from an acid solution using ethyl acetate. Glycine is an amino acid, and at pH 1 that amine would be fully protonated.
You could try adding HCl in either methanol or dioxane to your solution, calculated to just neutralize the KOH that you started with. The solubility of KCl in methanol is much less than the solubility of KOH in methanol, and most of it should precipitate. The when you strip down the methanol, you would be left with glycine contaminated with a small amount of KCl.
Is there a particular reason for carrying out this transformation? It seems like a lot of work to prepare something that is cheap and readily available.