I should clarify a few things.

-I already purchased new bulbs. The old ones worked but were very dim.

-The ballasts each measure about 30 ohms. I have not yet put an inductance meter on them but my guess is they are good.

-I connected the bulbs using jumper wires to eliminate bad connections in the sockets. No change.

-If I install only one bulb with everything wired as it originally was, that one always starts. If I remove it and install the other bulb in the other bulb socket pair, that one also always starts. With both in circuit, I've never seen them both start, only one does.

-With both bulbs in circuit one specific bulb tends to start, but randomly the other does instead.

-If I separate out each bulb-ballast circuit so they look like completely independent fixtures and start each manually by temporarily shorting the one bulb pin at each end, each starts and runs no problem. So that's why I believe both ballasts are good.

The part of the original circuit I'm having trouble dealing with is how the bulbs are permanently connected at the same end as the ballast connection. It seems one bulb could steal the start pulse from both ballasts, hence the reason for the issue. But then again this was designed this way and supposedly would work?

I have a DPDT relay on the way to isolate each starting switch and that will fix it, but did these desk lamps ever intend to work correctly? Seems odd..