How to Deep Clean Doug Fir Wood Floors Without Refinishing?

I have old doug fir floors which we had stained and poly'ed professionally when we moved in 10 years ago. Now they are marked all over, every which way, with black marks, scuffs, other markings, bits of dirt that won't wash off easily etc, etc. It seems regular damp-mopping won't do trick, huh? I can't afford to fully refinish them now. but wish there were a big deep-clean approach I could use, without having to refinish or wax them. I tried scrubbing a bit with a scrubby sponge, and found after 4 minutes, some of it budged in a small tiny spot....

I imagine getting down on hands and knees like old-fashioned image, scrubbing, but with what product(s?) and what material? Also read about tennis balls (could that be tough enough?), tiny-grade steel wool (but don't want to completely remove the finish, and we're talking about 3 rooms!). Maybe a little of each approach, in stages? What do you all recommend? Thanks so much! - (we have a special event here in 2 weeks). Thanks!

Lisa from , New Jersey