Learn to play with the timers. They are far from being that punishing, especially on lower difficulties where you only have to deal with 2 pods in the beginning of the game.
Then when the pods increase in number, you probably have a ranger who can scout them allowing for freer movements.
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I'm not saying this in a "get gud" kind of way, it's just that:
a - it's a turn based game, you can take your time thinking about your move.
b - It's completely balanced and designed so you have plenty of time.
If the timer keeps beating you, that means you are either not moving at all because distances are rarely that big or not using your resources efficiently enough to kill pods as soon as they appear.
If you are into mods, I wouldn't recommend to remove timers, but instead to use the "gotcha again" mod, which, among other UI things, helps you visualize where your specialist can be to hack the objective, being able to hack something from the maximum distance possible often shaves the turns necessary to do it by a significantly ammount. Knowing for sure that you can reach the target in 2 movements instead of 3 or 4 makes some missions basically have no timer at all.
You can be cautious, you just have to learn how. And again, it's a turn-based game, it's not about skill, it's about knowledge. You *can* learn how to deal with timers, you can learn how to move properly so you are always moving forwards while staying safe.
For example, as a rule, after concealement, you can do the first blue move with your fist unit to be as farther as possible, to cover as much ground as you need to approach the target, as long:
a) you stop behind any sort of cover
b) that unit becomes the imaginary line you cannot cross.
Which means that if your first unit of the round move the farther it can move and it does not activate a pod, you can rush all the other to the square directly behind it, forming a conga line.
And if the first unit does discover a pod, then you have an entire turn woth of actions to deal with it before it can hurt you, meaning that engagements should, at most, last 2 turns, 3 tops.
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On difficulties below legendary, you'll usually meet, at the start, one single pod before the objective and one single pod after. If the objective is a rescue mission, then it's one, then another, then reinforcements.
Still doesn't give you carte blanche to rush forward, but it means you won't be wasting turns fighting, meaning you can focus on moving the farthest possible with your first unit each turn. You can be cautious, you should, but you need to move forward.
Learn the timers, they are not there as a cheap difficulty spike to fuck players, it's there to create an interesting dynamic gameplay that rewards the player.