As far as i know they won't go away. If you want to get rid of them, trap them, burn them with lava or drown them with water.
For future reference attacking a slime with a weapon like a gold or diamond sword will cause it to instantly die rather than split. Its kind of like overkill.
Gold? Gold does equal damage as a wooden sword. You need at least a stone blade to one-shot slimes of Normal size. In order to do an insta-kill of a Slime, you need to do a half heart more damage than it has. Huge slimes have a TON of health (16 hearts!), so can't be one-shot by anything. Large slimes, their split, have 8, which also can't be one-shot. Diamond blades only do 5 Hearts of damage a strike, Iron does 4, Wood/Gold does 2, Stone does 3. However. When a Large splits into Normal slimes, Normals only have 2 hearts. So a stone, iron, or diamond sword
will one-shot a Slime at this stage. A gold/wood sword, on the other hand, will just cause it to split into Smalls which have exactly half a heart of health. A bare-handed slap will take them down easily, and since they can't actually hurt you are just there to look cute and hop around a little.
Or farm slimeballs, not that you actually need them for anything.
So in other words... Yes. You can one-shot slimes at lower tiers, but ya need a stone blade or better to do it.
EDIT: Small note, burning damage also means that they die without splitting. So using a clever little fire trap for them or soaking them in lava will assure that no little demon spawn come nibbling/sliming at your heels.