How do I verify that a 240v outlet is wired properly?
Given the 110V tester, I'll assume you are North American locale for this:

Look up your outlet type. 

Test (using appropriate care) that you have 240V from the L1 to L2 (hots) (you presumably did this per your question, but make sure it's the correct terminals)

Test that you have 120V from each hot to neutral.

Test that you have 120V from each hot to ground.

Test that you have no, or very low, voltage from neutral to ground.

Only after testing that voltage from neutral to ground is 0 or very low, turn OFF the breaker for this dryer outlet and test for continuity (resistance, Ohms) between neutral and ground - there should be continuity - the resistance should be low.

No need for a "widget" if you have a meter and can use it safely.

If the wiring is done very poorly you COULD have a situation where the voltages look fine until you turn on the dryer - that would be from something like a high-resitance connection - the voltage at the plug would drop significantly when the dryer was plugged in and turned on. Careful (always use care and respect with electricity) testing with a dryer in place could verify this. There's not really a safe way to test for this without a dryer to plug in. That sort of thing would be more likely if the outlet was installed by an amateur (I'm not clear if it was in place and you don't know it's provenance, or if it was just installed - if just installed, call back whoever installed it to check it out.) This is especially true if the wire is aluminum.