Commuting - Front or rear rack to reduce flex?

I've just started commuting and am currently borrowing a friends Trek hybrid with a rear rack while I decide what rack to use on my bike. I'm using an Ortlieb trunk case that clips on top of the rear rack and while the case is working out great the flex is not. I'm not certain if it's the frame, the rack, or both but the rear of the bike flexes horribly, so much that I cannot stand and pedal or the chain will just start skipping due to the rear end swinging left to right so much.

The fact that it causes the chain to jump between gears (it doesn't happen without the case mounted) makes me think it is just the whole rear end flexing. The pack weighs 20lbs, most of that is my laptop. I'll be switching to a Mercier Serpens next week and have considered multiple options, going to a seat post rack (seems like there would be even more movement in the rear end?), going to a stronger rear rack (would I flex all over with the carbon fiber seat stays?) or buying a fork with front rack mounts and going to a front mounted rack.

Can anyone point me in the right direction? Are any of these decent ideas or should I try something completely different? I thought about panniers but since 15 of my 20lbs is in my laptop I would end up unevenly weighting my bike, plus the single case works out very well when I'm off my bike, I just use it as a briefcase.

I am also unsure of what class of rack to get, I've seen the lighweight racks that are only good for 20lbs, all the way up to racks that are rated at 60lbs. I'm only carrying 20lbs so a lightweight rack would be nice, but will it end up flexing too much?

The rear triangle shouldn't flex that much.

Did a braze or weld come loose? Check closely!

Yeah, my old Voyageur was a very flexy bike but it still never ghost-shifted under power even when it had >30 pounds of groceries in the rear panniers and was fishtailing all that weight around. Check everything over.

If the weight's going on the rear, then I don't think you'll suffer much detriment if you use panniers and have the load somewhat unbalanced. Just strap the lappie in there so it doesn't get to throw its weight around.

Front lowrider panniers are good because they put weight on the front wheel, which has a stronger structure and an easier job, rather than your rear wheel which already has plenty to do. An off-center load up front does affect steering a bit, in my experience, but if the load isn't able to shift around, then I bet you could manage. Also, they put the weight very low to the ground where it has less effect when you rock the bike around.

I took my front rack and panniers off my touring bike the other day and LOL it felt actually squirrelly for a while, with all that mass missing down there ;)

Single panniers are pretty std for a lot of commuters. The only time it can be noticed is when climbing out of the saddle with a heavy load.
A rear rack is the usual mount and works well on most bikes with threaded eyelets. Look for a rack with 3 arms using Blackburn-style triangulation.
Any bike that is affordable will be striff and strong enough for a rear rack. Some weight-weenie racers are too flimsy but you shouldnt commute on them anyway.

Front racks are useful for balancing a touring load. They do slow down your steering response and the ability to jump obstacles.

It might be because all your 20 lbs is mounted pretty high on the bike. Have you tried a traditional pannier? With a high center of gravity, any standing pedalling will put lots of extra lateral stress on the bike.

It might also be that you have a worn drivetrain and the extra stress of moving 20 lbs is causing the chain to skip and/or the bottom bracket to flex. Remember that, with 20 lbs, you are essentially doubling the weight of the bike you are trying to accelerate.

I ride with my laptop in a computer backpack strapped to one side of the rack, rather than on top because it threw off the balance too much on top.

It only took a mile or so to get used to the weight on one side. After that it's no problem at all.

My biggest issue is that I don't have a touring rack... so stuff that flexes on the rack ends up in the back tire... I didn't see any touring type racks with the extra supports till after I bought it.

This means when I carry the laptop I have to use a strap passed across from the top front, and wrapped around the bottom rear corner of the backpack to support that corner and keep it out of the tire. This has worked flawlessly for about 4 weeks now!

I'll give it a try on the side of my rack instead of on top, the price is right on that fix if it does the job. Considerable drivetrain wear is a possibility, it's a ten year old bike that was used as a commuter for a couple of years by a coworker.