If I draw an exploding kitten while using the dig deeper card can I just put it back in the deck and keep the next card or do I need to immediately show the exploding kitten card and resolve it?
Reading the rules of this game it feels the intention is that you should be able to ignore exploding kitten drawn first but the use of the word 'draw' seems to suggest you don't.
From the rules here with emphasis mine
If you draw an Exploding Kitten and you don’t have a Zombie Kitten (or have one and choose not to play it), you are Dead.
So you die if you draw an exploding kitten. The rules clarification for Dig Deeper says
Draw the top card from the Draw Pile, look at it, and decide if you want to keep it. If so, put it in your hand. Otherwise you MUST draw the next card in the Draw Pile no matter what it is
Whilst it feels like this card should be able to skip the exploding kitten the use of 'draw' suggests not.
Compare this to rules for See The Future -
Privately view the top 3 cards from the Draw Pile and put them back in the same order. Don’t show the cards to the other players.
That card it explicit that you are not drawing, you are privately looking at card s.
If Dig Deeper said 'privately view' the top card and either draw that card or draw the next then I would say you could skip and exploding kitten. However the syntax of the rules does not say that so the exploding kitten would take effect.
However, to contradict this interpretation there is another note
This card is a great way to mess with the heads of other players. If you decide not to keep the top card, everyone will drive themselves crazy trying to figure out why you didn’t want it!
Looking at the rules Exploding Kittens seems to be the only bad card, I'm not sure why you would skip taking any others. It seems like a very badly worded rule if the intention is that you can ignore the drawn card but the syntax used infers if would still take effect.