How do I figure out the voltages at these 3 nodes of a circuit?
For a start you can forget all about the 1 k resistor until right at the end - its current is ONLY determined by the 6 volt source so that one is easy to find (if you need it and I don't think you do).

Then, try using superposition instead. In case you didn't know, with superposition you analyse each source on its own and produce a small table of currents. To do this you could start with the current source and the voltage source is replaced with a short circuit. If you wanted, you could start with the voltage source but then the current source is open-circuited for that part of the analysis.

It looks fairly simple if you start with the current source because then the 4 k and 2 k resistors become in parallel and also the 3 k and 6 k resistors become in parallel leading to a net impedance across the current source of 1.333 k in series with 2 k. This then tells you the voltage across the current source and from therein you can easily work out voltages and currents for each resistor.

Step and repeat by open circuiting the current source and reinstating the voltage source. Note all currents in both scenarios (sign is important) and finally sum currents for each resistor from both scenarios and you are very close to your answer.