The big issue with this seems to rely on being able to stockpile spell slots through short rests during downtime. Unfortunately, in the Resting rules (page 67 of the basic rules), it states:
Adventurers can take short rests in the midst of an adventuring day and a long rest to end the day.
This leads me to believe that RAW, characters cannot take short rests during Downtime days and so the theoretical character has no chance to stockpile spell slots.
The situation is neither normal nor abnormal
Every campaign is different. But it is normal for survival games to have nightly encounters.
That said, I played in an attrition-heavy campaign and we did not get nightly combats often; and even when we did, they did not interrupt our long rests.
Are you having fun?
Ask yourself and the players at your table if it's still fun. That is, is the survival aspect of the game ruining your fun? If it was taken away, would the experience be better?
If you and the rest of the players are having fun regardless, then nothing is wrong. If you are not, then you have to make it clearer to the DM that you are not.
Get on the same page
Since the DM is running a pre-made adventure, then make it clear to the DM that this one sucks and you are not enjoying it. Try and be calm and reasonable about it, maybe buy him beer too. What he doesn't need to know is the campaign is hard (he made the encounters, he knows this already). What he does need to know is that the players do not like the adventure.
If he is a sensible DM, he will adjust and ask you guys what you want in the game. Then, depending on if you are all able to agree on a type of game you want to play, he will either continue the campaign, modify it to fit everyone's interests, or stop the campaign.
If you do not like this particular experience, try and figure out what kind of experience you are looking for and be transparent about it. The DM needs to know this stuff so he can provide a better experience for you.
The rules on rest interruptions
A long rest is only interrupted if you do at least 1 hour of strenuous activity. You can fight for 59 minutes and still not have your rest interrupted.
Even if the rest was interrupted, you can immediately start it up again. The only rule for long rests is that you can only benefit from one once every 24 hours, not that you can attempt one once per dawn. So if you did not benefit from a rest within the last 24 hours, you can start one up any time.
Solutions from an in-universe perspective
While keeping everything about the game the same, you, as characters, have a lot of opportunity to influence the adventure.
Hire bodyguards while in town
- If you have NPCs follow the party for the purposes of protecting you, you can get away with nightly encounters without fighting the monsters. You also don't have to pay the bodyguards in money, but rather in promises, favors, blackmail, etc.
Stay in town longer
- Why are your characters adventuring away from the safe confines of civilization? Is there a pressing concern, or are you more "classical" adventurers who go out into the wilds for the sake of adventuring? If it's the second one, then you can spend a lot of time in town stocking up on money and supplies. This is money you can use to hire guards, buy scrolls, potions, etc.
Engage the towns so they can travel
- In survival games, the assumption is that the wilderness is unexplored and dangerous. That is why it's common to have nightly encounters. Then, your characters can build outposts between the towns, or they can engage the different towns to build these outposts that create a "safe route" for travelers. This will involve more RP than combat, though, which may not be something your DM is up for.
Establish outposts with defensive features
- When setting up camp, bring fortifications, set traps, put out the fire from your campfire (to not attract attention), create a decoy tent, create a decoy camp, etc. Get creative so that your rests aren't interrupted, and when they are, you can immediately go to safety while avoiding combat.
While in town, take a day to rest
- I think that no characters have died yet, from your question. So you can just take a long rest in town. If nobody has died in the party yet, you should consider that the DM is just doing a good job of challenging you as players while avoiding character deaths.
Ask the farmers
- As contributed by @GMJoe, ask the farmers and merchants how they travel. If there is any hint of trade at all between different towns, then they should have a way of moving between towns without being attacked every night. But as provided by @IllmariKaronen, just note that there might not be any farmers trading outside town at all.
Find someone who can cast Leomund's Tiny Hut
- As offered by @Nat, cast Leomund's Tiny Hut. This is a ritual spell that is available to Wizards and Bards, as well as anyone who has Ritual Caster, or the Book Pact/Invocation for Warlocks. This is a spell that guarantees a long rest and will stop nightly encounters from bothering you, at least until your DM starts using spellcasters against you.
Best Answer
In older (pre-4th) editions of D&D, there was a general philosophy that the purpose of the system was to simulate the functioning of a certain fantasy world. Thus magical powers would often be limited to once per day -- with the explanation that this is just how magic works -- but almost all non-magical powers could be used at will. The idea is that, if I could use an "awesome uppercut move" against the last orc, why can't I try the same thing again against this orc? There were also no "once per encounter" powers, even of the magical variety, because "encounter" isn't an in-world concept.
The 4th edition of the game changed this by adding once-per-encounter and once-per-day abilities for almost all classes. This was part of a general move in 4th edition away from simulationism and towards a more gamist view of role-playing games. For example, monsters in 4th edition don't obey the same rules as PC's, and the rules are very much designed for monsters to provide satisfying encounters with a minimum of fuss. This made things easier for the DM in most cases, but also led to the famous problem of, for example, orc swordsmen being unable to use bows or indeed any weapons not listed in their stat block. PC's in 4th edition had once-per-encounter abilities because it's a game, and that's how the game works.
The trouble with abandoning simulationism is that it undermines the reality of the world and the immersion of the players. If you know about once-per-encounter abilities but your character doesn't, it puts a barrier between your perspective and that of your character, since you're thinking about tactical considerations that your character doesn't know about. Instead of playing as your character during a battle, you're playing as yourself with your character as a game piece.
4th edition was criticized a lot for its abandoning of simulationism and embrace of game-oriented rules. Not all players disliked this change, but enough did that it became a controversial aspect of the new edition, and contributed to the split of the D&D player base into 4th edition players and 3.5/Pathfinder players.
Part of the goal of 5th edition is to re-unify the D&D player base by eliminating many of the more controversial aspects of 4th edition. For that reason, 5th edition has returned to a more simulationist vision of D&D, where the goal is to provide coherent rules for a world as opposed to rules for a game. That's why there's no once-per-encounter powers in 5th edition.
An escapist article based on an interview with D&D 5e designer Mike Mearls summed it up this way:
Regarding your question, there's nothing mechanically broken about adding such powers, and you should feel free to add them in your home game. You will have to adjudicate the question of what exactly constitutes an "encounter", and it might hurt some players' immersion and suspension of disbelief, but it might also help to make the game more fun and tactically interesting.