I purchased my new iPhone 5C Friday the 20th and the first day they were available. Since then, I've received texts and phone calls from people I don't know including a phone call from Walgreens about 'my prescription'. I can't get through to Verizon support by telephone. Of course I don't want to be charged for these phone calls and texts but they are also annoyances. I've already shared my new number with family and friends.
What can I do about it short of having my phone number changed?
You answered your own question. Change the number.
In many cases people have had spam annoyance calls, bill collector calls, prescription alerts, and if they had a load of friends calls and text messages. Its thed chance you take.
If you are the account owner you can go into My Verizon online and just change the number yourself.