Did you loosen the lower triple clamp bolts before adjusting the tension? I think that is necessary to allow the adjustment, or at least that is how it appears to my eye. Maybe our more experienced mechanics, like Mark Z or NJ Mike could comment.
Unfortunately, I've got more questions than answers here, but if Jess can fill in the gaps I'll be happy to try and help. In the meantime, here's my two cents for what it's worth.
I'm a little unclear on what's happening. are you saying the motorcycle becomes unstable while you're riding at a specific speed in a straight line, or does it only happen when you decelerate or accelerate?
Is the bike exhibiting a wobble, meaning the handlebars start to oscillate rapidly, especially when you're decelerating, or have a light grip on them? This is usually called a "tank slapper" for the obvious reason.
Or is the bike weaving, which is a slower oscillation that starts at the rear and makes the bike feel like it's got a hinge in the middle?
Normally wobble occurs at 35 to maybe 55 mph when you're either accelerating hard, and the front end gets light, which strikes me as somewhat unlikely given the RE's limited power, or when you're decelerating with a light grip on the bars. Wobbles are generally caused by mechanical issues, loose steering head bearings, out of balance front tires, loose spokes, etc. Typically, the riders grip is enough to damp out a wobble that occurs during deceleration, which is why riders that complain their bike "shakes it's head" when they take their hands off the bars are usually told simply to keep both hands on the grips. Wobbles generally disappear as speed increases, sport bikes and race bikes sometimes have a pronounced wobble when accelerating at full throttle from a low speed, most riders, at least the successful ones just ride thru it until everything calms down.
Weave is a bit more dangerous and much harder to control or resolve, in large part because every motorcycle ever built will weave at some speed, if you can get it to go fast enough.
Without knowing exactly what your bike is doing wobbling or weaving all I can do throw out a few suggestions most of which have already been mentioned.
First would be to check all the obvious stuff tire pressures, steering head bearing adjustment, swing arm pivot play etc. I'd also check any accessories like windscreens, saddle bags or anything else that's been added to the bike, make sure they're properly mounted and secure. Hell, even the mirrors can have an impact.
One concern I have here is that you changed one tire and not the other. Because the tires play a major role in damping out wobble and weave, a mismatch between tires can cause either problem, especially if there's a
profound weight difference between the two. Tire pressure is also a major factor, the higher the pressure the less damping ability the tire has, so make sure one tire or the other isn't overly inflated. There is also the possibility that your new rear tire, won't work with your bike, or won't work with your front tire. Some bikes are funny that way, which is why manufacturers often spec specific tires for their bikes.
As a general thought, the Interceptor/Conti's are generally stable platforms, but they do have a short wheelbase and a steep steering head without a lot of trail, although they're conservative by modern standards, their overall geometry would have been considered fairly radical not too long ago, so some chassis nervousness is to be expected, especially on the Conti with it's narrow clip=on handlebars.