MT3608/SX1308 boost converter and others

First of all lfb, due to the way AliExpress works it is best if you share non-localized links, that is, it is best if you share links which start with “www” instead of “pt” or whatever. When someone clicks/taps on a localized link (like the one you shared) the AliExpress site changes its language, forcing whoever taps on the localized link to re-select language and currency.

Onto the subject, the module you are using is not powerful enough to boost from such a low voltage to the amount of output power you need. The amount of power output is output voltage times output current, which is divided by the module's efficiency to get what it needs at its input: voltage times current. With 10+W of load and as low as 3 or even 2.5V of input voltage when battery state of charge is low, the input current at the module will go over 3 or even 4A (which that module cannot handle), this is easy to see.

You could check some DC/DC converter module reviews and get a good idea of how do they work or behave, like this one: Converter DC-DC 10A 8.5-48V to 10-50V 250W @ lygte-info.dk

Also, how are you connecting the battery to the module? If you are using some cheap battery tray, forget about it. If you need a high input current, you must minimize all input resistances. You need a proper way to connect the battery cells in that case: (low temperature) soldering or spot welding them. Bear in mind that when a current goes through some stuff/material, a voltage drops following Ohm's law (V = I × R), and power is dissipated (P = V × I = I² × R) too.

At last, let's not forget that you want to boost from low voltage, so let's find out a DC/DC module meant to handle these conditions:

A couple links:

As explained in the picture, the module lets you set input undervoltage cut-off/protection. For 2.5V, R22 = 12500 / (2.5 - 0.75) = 12500 / 1.75 = 7142.8̅5̅7̅1̅4̅2̅Ω, or -of course- the closest value you can find. In any case it defaults to 2V, which is enough.

A trimmer uses a screw head with multiple turns to allow precise adjustment, but as I said it is not meant to be adjusted frequently. Once you hear it click, this means it has reached its maximum or minimum end, and there's no reason to continue turning it that way. For this reason the lifespan of a trimmer may be rather sort if you do… exactly what you did. :facepalm:

On another matter, are you trying to feed a led strip using a DC/DC module? This is not the best way to power leds. Led strips consist in consecutive sections electrically connected in parallel, each one with a certain amount of leds in series and a ballast resistor. The ballast resistor drops some voltage and power, reducing efficiency, and is required to ensure that the leds don't burn out or overload the power supply. The reason for this is because leds' voltage drop or Vf diminishes as their temperature raises, causing even more current to flow through them if fed with a constant voltage power supply. A more proper way to drive leds is to use a constant current power supply or driver; the above module I linked can do this, and so you could try to feed leds without ballast resistor(s). The above linked constant current power supply or DC/DC module measures current flow internally (senses it in an ultra low value sense resistor) and adjusts the voltage output automatically to maintain the set current; current is set in the CC or constant current trimmer/potentiometer.

For the battery, I’ve soldered the wires in the little tabs that are spot welded in the battery. I was hoping to get something like 5W and not the 1W-2W that I’m getting.
When searching for xl6009 I’ve found this: “Wide input voltage range of 3V – 32V (optimum operating voltage range is 5 – 32V)”. So that must be the problem. The one you linked should work properly for 1s input. I’ll keep it in mind for a future project.
Anyway, I’ve bypassed the ballast resistors and added one more switch between the led strips, so I can, lets say, soft start and also have two brightness level. I don’t think that overheating will be a problem at that level of power, but thanks for the advice to use CC power supply.
I’ve made this video, so you can see the behavior that I was describing.