This could be incorrect fuelling due to a faulty water temp sensor. If this is the problem then the ECU will be leaving the car constantly ' on choke' and not reducing the fuelling as the engine warms - so unplug the sensor and clean the contacts on both sides with some switch / electrical cleaner (not WD40) and inspect the wiring for any signs of damage to the insulation especially where it runs over other components and through the bulkhead- if this doesn't help consider replacing the sensor as they're not expensive
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Ok if you now have transmission issues then I'd begin by draining off the oil and as it exits then strain it through some paper towel to catch any debris
this will show if there are mechanical issues with the unit
a wrong coolant sensor will effect fuelling very strongly and I'd also check if the fuel pressure is within limits as I've known the pressure regulator to fail and so overload the injectors
beoffre just replacing the injectors (as this can be quite expensive) I'd say its worth taking them to a injection specialist who can flow test them and also ultrasonically clean them