You all seem to be on target as far as this unit is concerned. I'll try to reply to most of your postings below. (Sorry for any redundancy.)
The sockets are made of plastic and most fit loosely on the base. Tightening might be impossible as there is a 1" specialized grommet that runs through the center of socket itself. I doubt I could replace them with a regular grommet tool as I would be squeezing the grommet through plastic.
There is also a screw on the bottom of the socket. I see wire fragments attached beneath it. Seems odd to me. Loosening or tightening the screw might be an option to explore. I don't think that screw is a supportive screw. I could be wrong. I haven't tried the eraser idea yet. There isn't any metal to bend up or down. The screw must be the metal connection.
The outer plastic sheath does rise rather high and may be causing problems in sockets I haven't tested. Will try removing each one that gives me trouble to find out if that might be a culprit. If so, I'll trim it down.
In some candles there is some play between the bulb and the socket.. Others fit snugly. I took the outer sheath off of one of them to see if that made a difference. Still had the same problem. There is too much play between the bulb and the socket. In other candles there is too much play at the base.
Thanks for the idea of replacing with porcelain parts or replacing the whole unit with a brass base lamp. Most likely buying a new unit is the route I will take. Although, I do have a friend who is electrically inclined. If I can get him to help me, I might replace each of the bulb units with porcelain.
Can't really replace with each bulb with 15 watt bulbs. This unit already gets pretty dim when 2 or 3 of the bulbs mysteriously decide to shut off. Has me thinking though. These bulbs are using a lot of electricity when the light is on. Maybe I should be looking at a more energy efficient lamp fixture. Or, if I do replace the sockets, then test out the 15 watt bulbs.
Thanks for your help.