Why does the voltage drop when connected to loads and what's the solution?
You've got a couple of things going on that are causing your problems:


The output resistance of TEG modules is fairly high.  They act like a battery with a big resistor in series.
By boosting the voltage, you increase the needed current.  Say you need 5V at 1A to charge your phone, and your TEG can deliver 2.5V.  Your TEG will have to provide 2.5V at 2A in order to deliver the 5 watts of power your charger needs.


Those two combine to cause your voltage drop.


Voltage always drops when current flows through a resistor.  The TEG doesn't have a seperate resistor in it.  Its construction involves materials and connections that raise the electrical resistance.  They must be made that way to work, it isn't an artificial limit.
You have the TEGs in series, so the internal resistance adds up.  If you have, say, three TEGs in series then you have three times the resistance.
Boosting the voltage increases the current draw from the TEG, and makes the voltage drop worse.


The problem boils down to your charger needing more power than the TEGs can provide.


You can try putting more TEGs in parallel.
You can try using a more efficient boost converter.
You could use larger TEGs.
You could put the TEGs in parallel and charge a low voltage battery, then charge your phone from the low voltage battery using a boost converter.


That last solution means you charge a low voltage battery slowly with your TEGs.  When that battery is fully charged, you use it with a boost converter to quickly charge your phone.