I noticed that when I slow down to 125 knots with only 7 tons of fuel on board, the plane sinks a lot and isn’t easy to control. Is this how the plane works? Or the flight model is off?
The 737 stall speed is 128 kts in the clean configuration
125kts is too slow for the 737.
Oh, I am flaps 40 coming to land.
Still approach speed is 140
What is the weight for that speed.
So it definitely will depend on which 737 you are flying, but here is a general operating manual for most of the variants you will have in IF.
It’s a lot to read, but it has some good information to reference. One thing to note is that it references a Vref speed. Vref is defined as " VREF usually is defined by the aircraft operating manual (AOM) and/or the quick reference handbook (QRH) as: 1.3 x stall speed with full landing flaps or with selected landing flaps." by flightsafety.org
Okay, I am flying the 737-700BBJ. It does have less pitch authority at my reference speed compare to my 777. However, in return, I can land on a shorter runway.
You are probably too slow. In general, on most aircraft, your nose should be slightly up on short final, not way up (which approaches a stall). The exception is the Q400 which approaches slightly nose down in flaps 35.