Switch the SIM, answer no when whatsapp asks you to switch numbers.
If you switch SIM AFTER installing WhatsApp, you are good to go with the old number. The servers "save" your activation phone number during that initial SMS.
I tested it on my own phone (it's a Gingerbread p1000n - All hail the venerable P1000) with a borrowed SIM card, the app just kept sending messages with my old phone number. WhatsApp did not ask to switch numbers. There is a "change number" setting.
If you reinstall the app while switching SIM, then you have to restart and get a new number.
About what is usually done when travelling, I can say from the World Cup visitors we received, several just bought a new temporary SIM card after arriving here in Brazil. The local phone companies even set up special and attractive prepaid plans for the football fans.
YMMV anyway, and you should check your home carrier overseas fees and the local fares. Usually you can get a better deal buying a trowaway SIM. Here in Brazil we even do it when traveling to another state.
If you want to keep your number and still enjoy the reduced local mobile internet fees, you can try and get any old android device that can do tethering (or buy a unlocked mobile hotspot), and then insert the other SIM into that device, then tether the internet to your main phone using the old phone as a mobile hotspot. Just turn off data and use wi-fi.
On Verizon's Global (not really global) plan, I urge you to read what Peter Shankman wrote of his personal experiences with that. My advice, is: DON'T. Or if you do, monitor closely your fees.