Blessing before taking

Hello,
I tried to look a little into the poskim who say that they used to wash their hands and only then recite the blessing (contrary to the opinion of the Rambam, for example). From what I have seen, the excuses revolve around the issue of busy hands. However, I did not find in the Talmud that busy hands prevent the blessing, but rather the eating of the teruma. Is there a source for this or is this unfounded halachic chatter and therefore perhaps one should follow the opinion of the Rambam and recite the blessing before?
Thanks in advance.