Actually Linux style is #10, not #13 (#13 is MacOS style, AFAIK). Also, note that it's #10 and not #$10 (which is #16).
The easiest way would be to replace the line ends on load/save, ie. instead of
Memo.Lines.LoadFromFile(FileName)
or
Memo.Lines.Text := STR;
do
uses System.IOUtils;
Memo.Lines.Text := TFile.ReadAllText(FileName,TEncoding.UTF8).Replace(#13#10,#13).Replace(#10,#13).Replace(#13,#13#10)
or
Memo.Lines.Text := STR.Replace(#13#10,#13).Replace(#10,#13).Replace(#13,#13#10)
and instead of
Memo.Lines.SaveToFile(FileName)
or
STR := Memo.Lines.Text
do
uses System.IOUtils;
TFile.WriteAllText(Memo.Lines.Text.Replace(#13#10,#13),TEncoding.UTF8)
or
STR := Memo.Lines.Text.Replace(#13#10,#13)
Of course, you should replace the TEncoding.UTF8 with the appropriate encoding you want to use.