The problem with a shotgun is that it is most effective at close range. Missions that require long range weapons will be difficult when limited to a shotgun. I'm thinking Luck of the Irish, Deconstruction for Beginners, Blood Brothers and Pegorino's Pride (and probably others). Having said that...Good Luck! If you get stuck on one of the missions let me know and I will try to help figure it out. I've played the missions all kind of ways in the past for the MKC!
My choice for a wrecking vehicle would be a Firetruck. It has speed (being an emergency vehicle) and good handling, it is a 4x4 and most importantly it has weight. Firetrucks are easy to find and if you set it on fire you can call for another one!
The disavantage of using a Firetruck is the water canon prevents being able to do a drive-by shooting (useful for popping tyres on your enemies vehicles)