In English ‘that’ can just about always be left out. And that is certainly the case here.
But it has been my experience that people tend to say “adamant that” or “adamant about.” I tend to say ‘adamant about.’
Any of these are acceptable:
I told her that she should stay home and rest, but she is adamant about coming.
I told her she should stay home and rest, but she is adamant about coming. (my preference)
I told her that she should stay home and rest, but she is adamant that she is coming.
I told her she should stay home and rest, but she is adamant she is coming.
Here, ‘to be adamant’ is used in much the same way as ‘to insist.’
I told her she should stay home and rest, but she insists (that) she is coming.
And ‘to be adamant about’ is used in much the same way as ‘to insist on.’
I told her she should stay home and rest, but she insists on coming. (my preference)
She has made up her mind. She insists on it. She is adamant about it.
These are all very similar in meaning, but vary in tone.
And going back to your first example:
Mr. Sanders has made up his mind about taking his campaign to the Democratic convention, but the primary election results are prompting him to ‘reassess’ his strategy, his campaign said.
Mr. Sanders insists on taking his campaign to the Democratic convention, but the primary election results are prompting him to ‘reassess’ his strategy, his campaign said.
Mr. Sanders is adamant about taking his campaign to the Democratic convention, but the primary election results are prompting him to ‘reassess’ his strategy, his campaign said.
These are close cousins to ‘stubborn,’ but as Prinz has already pointed out, not the same as ‘stubborn.’ Whereas ‘adamant’ is more situational – a reluctance to change after thoughtful consideration – stubborn is more of a characteristic of refusing to budge.
Mr. Sanders is being stubborn. He says he is taking his campaign to the Democratic convention. And the primary election results don’t seem to be affecting his strategy.