I would not use baby oil on a nice wood.
I have used linseed oil in the past for furniture. The finish looks good, but it is slow and takes a long time. The tradition is to use one coat per day, then one more per week, then...
The downside is that a traditional linseed oil finish may not be most resistant to water.
I use Watco natural Danish oil if I want a satin finish.
For a shine, I like Formby Tung Oil - it is really a wiping varnish, easy to use and can give you a good shine.
If I wanted a durable finish that is not quite as nice as the Watco alone or the wiping varnish, I'd use Watco followed by a light coat of wiping polyurethane (Watco). It looks OK.
You just need to decide what would look best and what sort of finish you want. Traditional oil finish - linseed oil - is susceptable to wear and is not as easy to apply as the others I mentioned. You can recoat Watco the same day, but linseed oil takes at least a day or longer between coats.
An alternative that is pretty nice is to mix linseed oil with a varnish (take your pick) and turps (1/3 for each) to make your own wiping varnish. I have used spar varnish for this in the past. If you take this route, stay away from any water based varnish for the mix.
If you don't have a good finish on the wood handle, you risk cracking with humidity changes in the winter and summer.