There is no setting for that.
SynEdit currently does not store line endings. It stores each line as a string in a list. The line ending is implicitly assumed.
When you retrieve the "text" then it adds the line endings after each line, including the last line.
So SynEdit.Text in your code always ends with a line ending.
You can add text to the last line:
SynEdit.Lines[Max(0, SynEdit.Lines.Count-1)] := SynEdit.Lines[Max(0, SynEdit.Lines.Count-1)] + newtext;
This is on the list of "maybe one day it will be added/changed". But that someday may still be rather far away.
Note that using SynEdit.Text or SynEdit.Lines is only recommended for read-only text.
Changing text this way will reset SynEdits undo buffer. So if the user can also edit the text, then the user looses the ability to undo.
If undo is needed then there is SynEdit.TextBetweenPoints