Some good advice. What I would do, and some of this will repeat the prior advice:
- Raise seat to the correct height. Place your heels on the pedals, and pedal backwards, the seat should be as high as it can be, your leg fully straight and just reaching the pedal at the bottom of the stroke, without your hips having to rock from side to side to reach the pedal. When the balls of your feet are on the pedals, your knees will be slightly bent and your ankle slightly flexed. This is very important. A low seat is very bad for knees. Imagine getting down in a crouch with your knees deeply bent, and trying to waddle for a mile, your knees would be in agony. Riding with a low seat is equally bad.
- Do not pedal with any significant force, at a cadence any lower than 80 rpm. It will feel like you are spinning your legs too fast. Get used to it. Pedaling with high force at a low rpm, "mashing", is another prime cause of knee pain. Pushing too much force through your knees hurts them, so instead push half the force, twice as often. You'd hurt yourself lifting 50 lb, but you could lift 25 lb twice no problem.
- When climbing, try to stay seated and spinning a low gear at 80 rpm or more. Standing pedaling is hard on the knees too. Same principle as above. You stand in order to push more force, but that is what you don't want to do. You may need to change the gearing. Find a new bike shop.

That should help a lot. But I think you should go further and use clipless pedals.
- With platform pedals, you can only push down, with your quads and that works your knees hard.
- With clipless pedals, you can pull back, using the hamstrings. You can pull up, with your hip flexors. You can push forward. Those are not as strong as pushing down, until you build up those muscles. But it takes some pressure off the quads and knees, and eventually you will "pedal in circles" and be much more powerful than when you only stomped down.
- Most types of clipless pedals have float, which protects your knees. For commuting, SPD work well. There are pedals with SPD on one side and platforms on the other.

By the way, you won't have to spin 80+ rpm forever. I do all those things - spin fast, mash slow, sit, stand, stomp down, pull back/up, etc. Your knees can take a certain amount of mashing/stomping. You need to know how to do the other stuff, to avoid forcing your knees to do more than they can. So force yourself to do the spinning/pulling thing exclusively until it comes as naturally as mashing/stomping, then you can start to mix it up.

Even if you didn't have knee issues, you'd want to learn to spin. When you exert force with your muscle, the energy stored in the muscle is used up. Your body can only replenish that energy at a certain rate. If you exert a lot of force and deplete faster than you replenish, rather quickly you will be exhausted. If you exert less force and deplete at a lower rate, so that you can replenish just as fast, you will be able to keep going and going for hours, as long as your cardio system keeps blood pumping and you keep eating. The saying is, "you want to ride with your heart and your lungs, not with your legs." For anything but short dashes, that is true.

By the way, it sounds like your bike might be too small for you, if there is not enough room between saddle and handlebar. The closer to horizontal your torso is, the less wind resistance, and when you get around 20 mph wind resistance is a big deal (or, riding into a headwind). Around 45 degrees is often a nice compromise between being aero and comfort/seeing traffic. If you can't get your torso down, maybe a longer stem will help. I agree with getting your bike fitted. By a good shop.