It's not about whether they stay longer, it's about cap space. The Patriots are basically up against the cap, and need some flexibility to contemplate signing a veteran TE or a veteran LT. Personally, I don't think they need to do either.
I'm in the camp that believes that the TE position is not critical to move the ball. It's good enough to have a TE to draw coverage in the Red Zone. The Patriots have plenty of skill players with diverse talent to move the football, particularly at running back with addition of Harris to Michel, White, Develin, and Burkhead.
LT is an entirely different story. Dante has small flashmob of tackles on the roster to start OTAs. While they might able to break into a rendition of Mama Mia to fill a train station, it's unclear that any one of them can be trusted with TFB's blind side. At 42, they cannot take a risk with Brady's health. They should be probing every opportunity for a reliable LT.