Does a TIP120 transistor have an internal resistance when switched on?
I'm pretty sure the transistor is getting switched on all the way


The TIP120 is a Darlington transistor and NEVER can get switched "all the way". It will always "drop" at least 0.5 volts across it and reduce the voltage across your solenoid.



If you think about the picture above, for the right hand transistor to be "on" there must be about 0.7 volts at its base with respect to its emitter. That 0.7 volts is sourced via the left hand transistor from the collector pin hence, the collector can never be lower than about 0.7 volts and the right hand transistor can never be fully turned "on".

Here's an extract from the data sheet that shows what I mean: -



At 240 mA collector current \$V_{CE(SAT)}\$ is about 0.75 volts. And this assumes that your base driving circuit is delivering a current of about 1 mA.