Viruses do have a tendency to recombine, as does any organism with a rapid rate of reproduction like bacteria etc.
Live vaccines are always risky for this reason, there are many cases of live attenuated vaccines recombining with the wild type and causing disease instead of preventing it. There is a theory that the variant of bluetongue virus circulating in Italy is not actually from midges, but from their use of an African strain live attenuated vaccine which caused disease in the 'wimpier' European breeds.
I'm not sure that public health officials need to talk to evolutionary biologists more, unless these public health officials are vets or scientists. But the researchers developing new vaccines, particularly those working for private firms where profit is the bottom line, do need to keep their eye on the ball and keep recombination in mind when developing new vaccines.
Live attenuated are never the first port of call for vaccines, for precisely this reason. Vaccines using a part of the virus shell, other virus proteins or, the hot topic of the moment, viral DNA, are more popular because it's just a tiny part of the virus and less likely to result in any direct effect on the wild type strain. But nature is a fickle beast, and anything that can happen probably will. Researchers never know what effect adding anything to any biological system will have. That's why very stringent controls are in place before anything is allowed out of the lab and anywhere near the real world