Weak Brakes

My old 1997 Dodge Caravan (150k miles) has weak brakes. During normal driving, it needs more pressure on the pedal to brake normally, than any other car that I have driven. This would be ok as I can push hard on the pedal... But the problem is that, past some point, I get no additional braking power no matter how hard I push. So in case of an emergency, I cannot come anywhere close to lock the wheels (and this car does not have ABS). So basically, the braking power is proportional to how hard I push, up to a point where the braking power remains the same no matter how hard I push. At that point the pedal is maybe 1/3 down, still very far from the floor.
I have just replaced the front pads and rotor with no improvement. And when the engine is not running, the pedal gets really hard after I depress the brakes 3 times, indicating that the booster is doing something.
The car will stay in the driveway for now as this is a dangerous situation. Any suggestion?
Thanks,

your brake booster is out, or a bad vac leak in that line.
or 2 your rear brake cylinders are leaking resulting in loss of pressure unless you \push hard to overcome the leak

Things to check for, loose or leaking vacuum hose to booster, calipers binding or dragging, caliper, wheel cylinder or master cylinder pistons sticking or seized, bad vacuum booster, master cylinder compensation ports blocked, brake hoses, lines or fittings clogged or restricted, low engine vacuum or bent, broken or dragging rear brake shoes.

Thanks for your ideas. Trying to eliminate some possibilities:
-2 rear cylinders leaking: Unlikely as this problem has been going on for a long time and the brake fluid level has not dropped
- Bad vac leak: The line from the manifold to the brake booster is visually ok. There are several other lines attached to the same vacuum port. I’m going to attach a vacuum gauge to confirm that vacuum is ok.
- Brake Booster bad: After stopping the engine, the pedal is softer the first 3 times that I press it, and then it becomes real hard. Is there a good way to check/diagnose the brake booster?
- Caliper sticking or seized: When replacing the brake pads, I had no problem retracting the calipers with a C-clamp. The problem was there before and after replacing the front brake pads.
- Master cylinder pistons sticking or seized: Any suggestions how to test this?
- Master cylinder compensation ports blocked: Any suggestions how to test this?
- Brake hoses, lines or fittings clogged or restricted: Visually, from the outside, the line and hoses look ok. If clogged or restricted I figure that the hydraulic pressure would still eventually build up if I hold the pedal down for a while. If a line was plugged, then nothing would come out when I bleed the brakes
Thanks

also, how long has the fluid been in there? it may be so old it has almost no service life left.

Yea, the brake fluid is old and I am planning to flush it. But, if I understand correctly, old fluid may contain some water and would not work as well when the brakes are very hot but would still work ok under light/normal condition.

I'm trying to decide if and what part I should order...

By the way, the engine vacuum measured at the manifold port is 18 in hg at idle, which is normal. Then, if I depress the brake pedal, it remain the same. I don't know if this is normal or not.

Thanks

Check the brake booster by stepping on the pedal several times with the engine off like you did. Then still holding the pedal down start the engine. If the pedal sinks further as it starts then the booster is ok.
Pull the rear drums and see if anything is out of whack there. Glazed or contaminated linings, poor shoe/ drum contact due to worn out drums, broken springs,etc. Outside of that I'd say the master cylinder is bad.