Hi, I'll bet we all had wrinkles and puckers on the back of our early quilts. I know my first ones have. They are still loved and used.
You don't need a new machine. I only sew on vintage singer machines, most from the 1950's. A walking foot will certainly help, and they fit the old machines.
Enjoy your stitching.

My machine is a 12 stitch Kenmore (serial number 385.1274180 if that helps).

Those Kenmores are awesome workhorses, and you should be able to decrease the pressure on the presser foot. It's probably a pop up and down silver button above the needle on the top of the machine. I agree with the others that you'll probably have better luck in the future with spray basting.

Welcome to the Board. You will learn a lot here. I was told you don't give away your first quilt. Would you be happier to keep this one for yourself and make another to give them next year? I sew on a 1967 and a 1949 Singer, there is a whole section on here about vintage machines because a lot of people love them. I have just finished my third quilt and given it away. I sewed from the centre across each sashing and when I started to sew up and down I found that I often had a little tuck as I came to the edge of the next block. I folded each tuck neatly under and washed and sort of half ironed it. I don't think he will ever see the tucks. It would take someone deliberately and closely looking to see them now that it is washed and crinkly.

Welcome to this board. Lots of helpful, knowledgeable people always willing to help out.

After me quilting 4-5 hundred quilts I still sometimes get puckers on the back. One I unpinned a puckered spot and re pinned it, still puckered. Just a few. It is for a grown Grandson he probably won't even notice. He has several quilts from me. I only use large safety pins.

my husband's favorite quilt was one of my firsts, but all the straight line quilting I did ended up turning into V's at the intersections with puckers on the top and bottom. that being said, don't rip it out (I think you will end up with a distaste for quilting if you rip it out), but know that it can and will be loved and used plenty even with the mistakes.

Welcome to the QB. We've all done our share of 'screwing up', but that's part of learning. Others can instruct you on what and how to do the quilting on your sewing machine. My back/arms/hands don't allow me to do that, so I am clueless about it.

Don't give up. Like I said, we've all screwed up with quilting. Learn from it, and continue on your quilting journey.

Lots of good suggestions here. You might also try removing the presser foot and making sure it is smooth and clean on the bottom. Sometimes older machines get a bit gunky and that could exacerbate the problem.

All good advice, but I didn't see anything mentioned about when you're starting the "sandwiching" to have your backing taut, but not too taut. I tape my backings to my cutting table then continue with the batting and top. I absolutely HATE pin basting, I do like the spray basting but rarely use it because of the "overspray", so I thread baste my quilts.

In looking at photo images of your Kenmore, I'd say it's a more recent model than the one with which I started quilting. I was able to buy a generic walking foot for mine, so I would think you should be able to get one, too. Look at this page on Amazon: https://www.amazon.com/Deluxe-Walkin.../dp/B01AEC5PNK