The bar tends to slip down until my fingers are like hooks and the bar is barely touching my palm. Should I be gripping tighter?
You need to strengthen your grip. Eventually when getting near 1RM (usually around 3RM) most people will need to flip to a "mixed grip", where one grip is overhand and the other is underhand. But it's a bad way to train and studies show that it doesn't develop you symmetrically (can't find the study at the moment).
Pullups, chin ups, farmer's walks, and heavy kettlebell work is a great way to start. Deadlifting works great too.
Answering your question directly, yes, you should be gripping tighter. The stronger your grip gets, the easier it will be.
If the bar is slipping from your grip that's generally a sign that you need to grip the bar better.
That could mean gripping tighter, which is always helpful, but it could also mean using chalk or, if the weight is heavy enough to justify it, a hook grip.
As everyone already mentioned, your grip is fatiguing sooner than you complete a full rep.
Static bar holds, pull-up bar hangs, heavy deadlifts strapless, plate pinches.
Mixed grip causes some imbalances in muscle development if you are over-reliant on them. Swithching between that and the double overhand will give you the best of both worlds.