You can use a regular Multimeter to check continuity but you need to pay close attention to what its reading. A set of closed contacts should be a dead short. It should read virtually the same as if you touched the two test leads together. Same goes for checking a fuse (unless it is rated less than 1/2 amp). On many meters, that would be something like 0.4 ohms. A typical DVM would have a low ohms setting of 200 which may or may not give you readings less than 1 ohm. 2 or 3 ohms across a set of contacts would probably be a "Bad" reading since some solenoids have a load of less than 2 ohms. This is why even though a contact appears to be closing, it may not be making good electrical contact for an EM logic circuit. Many of the newer DVM are auto ranging and may or may not go down low enough, or allow you to "lock" them into the lower ohms scale. If it can't lock, then each time you test, you have to wait for the meter to start in the high scale (megohms) and work its way down to the lower ohms scaling before you will get a valid reading. It has been so long since I have used an Analog meter (needle style) I am not sure how low they go to read low ohms readings but should go down to a 10 or 100 ohms scale if I remember correctly.
If you have an auto parts store near by, you can purchase a points file (or a needle file set at a local hardware store) to file your points. You don't have to wait for your flex stone to arrive. If you go with the metal file, always turn the power off when cleaning contacts. These are my preferred tool over a flex stone.