Yes, each one tests okay with 240 when testing both poles.
The only thing that I can come up with is that the breaker is bad, but I don't understand why I get the 120v when testing each pole seperate.
That depends where you are measuring. If you measure from each hot leg to ground or the Neutral. You will measure 120 Volts. You said you replaced the receptacle with a 4 prong unit to match the bew Dryer's plug? Is your cable from the panel to the dryer receptacle 10/2 plus a ground or is it 10/3 plus ground. the proper cable should have a red, a black, a white and a ground wire either bare or green to properly connect the 4 prong receptacle, and at the panel end, the red and the black should each go to one phase of that 30 amp breaker. the white should connect to the bus strip with the other white (neutrals, and the ground should be connected to the ground bus. The ground and neutral buses are seperate, but may be connected together by a bolt in the panel. In any case, you first need to measure the voltage at that 30 amp breaker at each phase to make sure you have 120 volts to ground and 240 (or whatever your actual 2xx voltage is in your area) between the 2 phases. The breaker may be bad, but if it worked before with your old dryer, I am under the opinion from what you wrote so far, you may have wired something wrong at the new receptacle. Just my opinion, and you know what opinions are worth.
I reread your posts and just noticed you installed the new breaker (was there one there before? Is the new breaker plugged onto both metal "stabs", one phase on each of the stabs?. My advice sticks: first measure at the breaker itself, if you have not already done so.
Everybody else that has answered you is correct in what they said too, but we need to get all the details of what you did to figure out what is wrong. It is possible, but not very likely that the "new" breaker you installed is bad.