It does not mean baby will come sooner, no. Baby can be engaged for weeks. My firstborn was engaged for weeks whereas my third would NOT engage until I was actually in labor, past my due date.
Bad leg cramps is normal in pregnancy. One tip when you get a "Charlie Horse" is to flex your foot, don't point it! Flexing your foot (imagine pointing your toes toward your forehead rather than pointing them like a ballerina) will make the pain stop almost immediately. Then drink tons of water because hydration is key to preventing them.
I also have more back pain with each pregnancy. I'm pregnant with my fourth daughter and my back, hips, groin, EVERYTHING hurt. Hang in there! You're almost to the finish line (and I'm only halfway and in pain!).
With my first I never felt a contraction until my water broke and I was 4+cm. With my second EVERYTHING was so much more sensitive. I had contractions for months. She was at a -2 station (which is almost fully engaged) from week 32 on. I was also 3 cm and 90% effaced. I didn't give birth until 39 weeks. Honestly baby is going to come when it wants to. The only major sign for me was my bloody show. I had that and delivered in less than 24 hours.