The following things may be more than you want to do.
First you can clean the tuning capacitor per Dr. Audio's post. This may help with the sensitivity and it is a good first step.
1. Vacuum out all the dust you can from the interior. Don't touch the fins on the tuning cap!
2. Blow the dust out of the fins with compressed air. Don't touch the fins.
3. If you examine the tuner cap, you will see brass contacts at each point where the shaft passes through the metal housing. (Including where it passes through the internal walls.) These make the ground contact and they get dirty. Buy some NON - RESIDUE contact cleaner, and inject it into all these brass contacts, then work the tuner back and forth across the dial at least 20 - 30 times.
4. Then apply just a drop of CAIG Faderlube to each contact. It is better to use a syringe to apply it, rather than trying to spray a small amount. If you contaminate the fins of the cap or the circuitry, you will detune the circuit. Work the tuner back and forth 20 - 30 times again.
This procedure will cure all kinds of woes; noise when you turn the dial, certain stations not being received sometimes, when other times they come in clearly, etc.
See the picture below. Next, you can use your DMM to measure the voltage
across resistor R125, a 300 Ohm resistor. In the FM mode, this voltage goes directly to the tuning meter through the mode switch. If you measure the voltage here, your are measuring the voltage before the switch.
If you connect the positive lead of your DMM to the point circled in red and the negative lead to the point circled in blue, when start to tune in a station from below its frequency, your DMM should show a small negative voltage and as you approach having the station tuned in the voltage should go to zero and as you continue to tune up in frequency the voltage should go positive.
In the next picture I have circled some adjustment points having to do with the sensitivity of the turner and the position of the meter when in the FM mode, but in is not usual for these to be far enough out of adjustment to cause you problem unless someone has changed there adjustment, although sometimes the trimmers TC2 and TC3 can be a little sketchy.
You do not want to clean them, but you can use a Sharpie maker and mark their positions by making a mark on what looks the the top of the screw and making a line from there to the body of the tuning capacitor. You can then work them back them back and forth several times, then return them to their original position by lining up the marks. This may or may not help (I have seen it help) but it is easy to do and is part of the process of elimination.
If you connect you DMM to TP1 and ground as shown, you can measure the received signal strength. It should be around at least 2 to 3 volts DC or more on a moderately strong station. This can be an indication of the sensitivity of your receiver.
Transformer T2 is an inductor and will likely not benefit from moving its adjustment back and forth so you can ignore it for now.
I will stop at this point because I do not know if you are comfortable with the above. And if not that is okay, not everyone wants to dive into this level of trouble shooting a tuner and that is quite understandable.
There are only three adjustments for the FM sensitivity and one for the center tuning meter. As I have mentioned these do not usually go out of adjust far enough to cause your problem unless someone has been in the tuner section. I can post those if you are interested. They use the same test point indicated above, using your DMM with your receiver tuned to a stations at the upper end of the dial.
Sorry for the lack of pictures, AK has gone sketchy on me, I can not post attachments and I have lost all of the options that are normally at the top of the window. I will try to post the pictures that I referenced at a later time.