Ordinary lemon or orange oil would be fine on it. Use it on a soft cloth.

I did not know you could put oil on rocking chairs

dadchin, I grew up where you are now, and oiling wood makes a huge difference in furniture lifetimes, as well as appearance.

Use lemon oil, as someone already suggested, and consider having those dowels reglued. Keeping the wood moist may restore their tightness, but it sounds like you could be a little too late.

I used to try to oil about once a month, although I probably missed it once in a while.

The best way to care for clear varnished furniture is with a coating of good paste wax. Good quality paste wax is available at most hardware stores. Do not wax furniture that is painted or has a flaking finish.

http://www.loc.…

Or use orange oil.

http://www.touc…

Hope these methods work for you!

Orange oil does seem to be recommended the most.
Here's a manufacturer that I highly recommend - General Finishes.
This link will take you to their furniture care and maintenance page -
http://www.gene…
Here's their store locator page - http://www.gene…

And here's where you can get it online - http://www.rock…

Liquid Gold is awesome and was recommended by furniture builders that were close friends until we moved.

Another solution is making your own recipe for furniture polish that I got from http://www.abou…

1 cup olive oil

1/2 cup lemon juice

Mix together in a clean new spray bottle. To use, remember to shake before each application. Apply a small portion to your cleaning cloth. Spread the polish over the furniture, trying to polish evenly. Use another clean cloth to polish the surface dry.