How to cope with bad rolls on important actions?

Talk to your DM about making failure awesome.

The priority in any RPG is that it should be fun first, storytelling second and mechanics a distant third. D&D 5e makes this explicit in its how to play section (PHB p.6):

1. The DM describes the environment.
2. The players describe what they want to do.
3. The DM narrates the results of the adventurers’ actions.

Of particular note is:

the DM decides what happens, often relying on the roll of a die to determine the results of an action.

Often not always!

Before calling for a die roll there needs to be a clear difference between success and failure and both of them should advance the game! If there is no difference or failure is just boring don't roll the dice; the character just does what they want to do. However, it is much better if failure is at least as interesting as success and leads to a new environment where the players have new options.

Specifics:

we decided it might be a good idea to let me sneak near the camp to look for guards and count how many people there actually are.

This is what you want to do - success means you get this information, failure means you don't. Both of these are interesting in that they allow the players to make different choices, in one circumstance with adequate information, in the other without.

The failure state should not (necessarily) involve you in an unwanted combat.

This is how it goes down at my table:

You are creeping up on the camp when a stick beneath you cracks with a loud snap! "Who goes there?" shouts a guard "Charlie, there's someone creeping around out there, get the sergeant!"

You haven't been seen yet but the guard is now on high alert. You can quietly back away or press on. If you press on you will be making your Dexterity (Stealth) check against his active Wisdom (Perception) with advantage.

See how much more interesting this is than: "He sees you, roll initiative." You still have control of the situation (which is as it should be) and you can take a big risk for success or back away and accept failure.

What I've personally witnessed in the last two weeks:

• Throw your dice clear across the room.
• Talk soothingly to them and reassure them that it's not their fault.
• Make pointed comments about replacing them with the other bag of dice you have in your bag.
• Threaten then with microwave time.

In my experience, comments and threats don't work. You have to follow through. Line up all your dice on a concrete step so they can all watch you use a hammer to smash the worst performer into coloured plastic dust. :-)

Seriously though, we've all been there. Sometimes the random number generators just aren't with you. Don't sweat it, just keep playing.

The primary mechanic 5th edition offers to help with crucial rolls is Inspiration.

The halfling racial feature Lucky, the divination wizard's level 2 Portent feature, and the Lucky feat are additional ways to get around poor rolls when it really matters.

For you: Get new dice.

Dice are not perfectly balanced and over time they do wear away, the cjeaper ones faster than most. You don't end up rolling a 1 every time but the probability can be skewed to lower or higher numbers.

A test of this is to put a dice in a glass of water and tap it, see what number it comes to and repeat. Plenty of videos on google of this. It is a well known cheating trick to create weighted dice, its possible yours are weighted negligently.

I personally get new dice for a new campaign, not necessary but who doesn't like shiny new dice :)

For your GM: Use extreme rolls to add tension

When I GM I treat natural 1 and 20 as epic rolls, something epic happens not double damage.

In the scenario you mention I would have had the bandits critical shot shoot the bow from your hand and scatter it D20 feet from you. Yes your unarmed but your alive, now you have a to make a choice, draw a blade and go melee or ... pull an arrow from your quiver, dive for the bow thinking this is your moment and take that all or nothing shot.

Maybe you miss yet again and he shoots you dead while your prone on the floor but at least its a cool death. Now you can laugh with your friends about you ridiculous luck instead of feeling jaded about that critical.

I once had a player with the same issue as you, after the second session of this the party decided he was cursed and dragged him off to the local cleric who confirmed he had in fact offended the gods, cue side quest of atonement.