Did any Nazis sneak into foreign countries among refugees fleeing the War and/or Holocaust? If so, what did they do?

Not in America, they walked in settled down and lived pretty good until they got into with their neighbor and the neighbor turn them in.

Yes.

In fact, quite a few SS troops from the Eastern Front joined the French Foreign Legion.  The FFL used them in VietNam, until France gave up that mess and the US tried its hand there.

Of course, most of their scientists were taken in by either the US or USSR.

Many leading Nazis were assisted in their escape by the Odessa network and, of those, many settled in South America where they were protected by sympathetic regimes.

But the smaller fry often made their way to the Allied countries where they settled down secretly and, for the most part, lived out their lives like anybody else.

My next-door neighbour for almost 30 years was such a man. He had been in the Hitler Youth during the 1930s, joined the Luftwaffe during WWII, and emigrated to Australia during the chaos that followed the wear.

He died, an old man, in the 1990s, but until then he still maintained his love for Hitler and his hatred of the Jews. Old Nazis didn't suddenly become decent people, they just learned to cover up most of their beliefs.