I have tried to jiggle the two parts and they are not joined at the top or sides; just at the bottom. I can't figure out where to put oil since there's no room under the buttons.
Is there anything else I can do or is this a common problem that is more serious than that they just need cleaning and oil?

Nona, March 2013

Hi Nona,
I guess you mean the two buttons in the centre of the carriage?
They do get stuck together and it helps to clean them with a brush dipped in surgical spirit. This removes the old oil and fibres, then you can apply fresh oil. Check the underneath of the carriage as well as it may need a thorough clean and oil.
This should help rectify the problem.
Regards,

Sue, March 2013