Continuity mistake: When Donovan turns into a living corpse the badge on his lapel disappears.
Continuity mistake: When Donovan drinks from the grail his left hand holds the base with extended fingers. A frame later they are bent.
Continuity mistake: When Indiana's dad blows up the truck beside them during the chase, there's a big explosion. However, in the next shot showing Indiana and Ernst Vogel, the same explosion behind them is smaller.
Continuity mistake: When Indy and Kazim are about to jump off from the chopped boat, the boat swaps from sinking to floating between frames.
Continuity mistake: The position of the stone that Indy places to block the cannon changes between shots.
Continuity mistake: Right before the castle is on fire, there is a chair very close to the fireplace. When the place starts to burn, father and son take shelter under the fireplace thanks to the fact that the chair has magically moved several meters away.
Continuity mistake: When young Indy says, "Cortés gave away the cross in 1920.", the thief is holding it with both hands. A frame later he holds it with just one hand.
Continuity mistake: When Indy and Elsa come up through the manhole their hair is sopping wet and messy. As they run to the boat their hair is magically styled and not sopping wet anymore.
Answer: They're not, no. An individual doesn't become immortal after one drink - it requires them to drink regularly in order to remain alive. So neither Jones has been rendered immortal, merely healed of any wounds that they might have. But your surmise is basically correct - as the Grail cannot leave the shrine, any individual wishing to use it to prolong their life must stay there if they wish to enjoy its effects.
Also, in Indiana Jones and the Kingdom of Crystal Skull, Henry Jones, Sr has passed away before the start of the story and therefore was not immortal.