They're not states they're colonies
Your scenario is very close to that of the European powers colonising as much of the world as possible some 500-600ish years ago. The challenges are very similar as well, cargo and people move slowly between motherland and the colonies while information though limited can travel a lot faster. This is a space-age version of those exact same problems.
The remote settlements won't be states, it's simply not practical, again look to European colonial history. The remote settlements will be "territories", "possessions" or colonies. (I'm not sure if there's any legal distinction between the different terms, be interested to find out). They may have some small influence on the politics of the motherland but they won't have equal status. The British electoral system is a good model for how things would almost certainly run.
In that model, colonial citizens have the right to vote in the motherlands elections if they were resident in the motherland at the time the election was called. MOtherland citizens have the right to vote no matter where they're living at the time.
An excellent example of how your scenarios would play out is the Coyote Series
Written by Allen Steele, the original trilogy was a great read for me back-in-the-day. The first spin off novel, Spindrift, didn't have the same sort of epic vista as the first three so I left the series after that, but the first three are right up your alley.
If your FTL data channel is limited to, say, 24bps, what's to stop them opening up multiple channels ? If that's allowable then the obvious next step would be to bond hundreds of the 24bps channels into virtual channels with more bandwidth.
Even restricted to 24bps the data link would be useful. Allowing for error correction and assuming its duplex, you'd get about 16bps data, or two ASCII characters per second. 120 characters per minute or about 15 words. Which isn't great, in the 19th century, human telegraph operators managed an average of 30 words pm and ticker-tape machines went up to about 70s word pm.
In reality your throughput would be much better. It would be a digital not an analogue connection, it would be running 24/7 and data compression methods would help your throughput enormously. It would be limited to text only but you'd still get a respectable throughput.
But the citizens won't see it like that, their experience will be much worse. Even though the actual data is travelling quickly the amount of data is limited, so the "telegraph office" in the colonies will have to impose a queuing system, and of course private citizens personal letters back home will have a lower priority than official comms. So from the citizens POV it still takes weeks to get even a short post-card length reply from home. I imagine that would cause some resentment.