It is practically impossible to avoid any burr and it is impossible to avoid the thread tapering down to a thin feather as it reaches the end of the stock.
To reduce the burr as much as possible you need a chamfer the end at the same angle as the thread flank.
You can chamfer before cutting the thread but you will still get some burr, however if you go back and re-chamfer and then take another final pass down the thread you can minimise this.
One way to do it without tool changes is to use the threading tool to chamfer and make the tool path go down the chamfer starting at the largest diameter.
If you really want to get rid of the wavy edge altogether you need to find out about a 'Higbee' cut. This completely removes the wavy edge by using a straight tool to cut off the thin part of the start and then pull out very quickly leaving an abrupt start to the thread. Try doing a search for Higbee and you should find something.
An open mind is a virtue...so long as all the common sense has not leaked out.