Every sidecar is a custom job, no two are ever exactly alike even using the same bike, i.e. an R3. They all have their "headshake", at about 26-32 mph. some more than others. We just power thru it. If is a more serious shake, there is no one solution, it can range from adding tire pressure to steering damper to alignment. Start with the least expensive solutions, like adding tire pressure and work your way up the line.
Try to find an experienced sidecar owner in your area that can help you sort it out. Your being new to sidecars it can be intimidating at first because of all the new variables involved in dialing it in. Since the previous owners put a lot of miles on it, I would assume it was set up okay. Learning to pilot a sidecar is not difficult, it's just different. You will need to unlearn some riding habits, like diving into curves and corners. Do that and your sidecar will be off the road, bouncing against curbs, rocks and potholes causing you to lose control.
When accelerating the rig will pull to the right, to the left when braking. This is normal. Taking a right turn too fast and the chair will "fly" on you. Not good, even dangerous, until you practice and learn to control the flying. During practice on a large empty parking lot, intentionally make the chair fly to get use to the feeling. Then you won't panic when it happens on the road. Many add ballast to the chair to keep it from flying when riding without a passenger. A couple cases of bottled water will do the trick. Never ride with two adults on the bike with the chair empty or with only a small child in it. You want to keep your weight more balanced between the two. Also don't put heavy weight in the nose of the sidecar, is safest over the axle.
Get the yellow book if you haven't already
Riding a Sidecar Outfit - Must have sidecar training manual | DMC Sidecars
It has tips, hints and exercises to practice to make you a safer sidecar pilot.
Several have added steel plates to the sidecar frame as added weight. The R3 can handle any weight you want to put there.
Stay with the progressive front springs, changing springs in the R3 forks is a major undertaking, requiring special tools. The progressives are much better than the stock springs.
Many go with a car battery for more juice when turning over the engine. Carrying that larger battery is no problem for a sidecar, often is mounted between the bike and the sidecar. With the R3, it is documented that running a larger or additional ground wire to the engine helps with starting. As Turbo suggested, the Toyota 2.4 starter is an easy fit and much less expensive than the Triumph starter.
That's my two cents for today. Be glad to answer any sidecar question for you.