Typically, I believe, if you decelerate with closed throttles, while holding the bars loosely and you get a shake, this would indicate excessive play in the steering head bearings.

Oops! …. Congratulations on your first Airhead!  Welcome!!

Congratulations and Welcome Aboard !

In my experience, wobble is always caused by something wrong in the "front end", aka "the forks". Get the front wheel off the ground and I'll offer a whole list of tasks in relative order of importance. However, with so much history unknown to you, I'd be doing all of these. Maybe not all at once, but at a minimum within the first 2 weeks...

• Design. IF your tire has a center groove or central rib (the OEM Continental ribs for example), then that feature in the rubber MUST run perfectly true. Hold a pencil against the front fender and spin the wheel. Because this tire design will follow "rain groves" in the highway, these tires are not seen much anymore. If you need to buy a tire, be sure and purchase one with a block pattern and NO central feature. [Also, the Dunlop K81 is famous for being poorly molded and has a center groove that is not true.]
• Mounting. If the central feature does wander, is it the tire or the rim? If tire lube is not used during tire mounting, then the "bead" of the tire may not "pop out" and fully seat the tire correctly on the rim. There is a small molded-in line in the sidewall that allows you to feel if the tire is fully "popped out". Set your finger nail against the rim and slowly rotate the wheel while allowing the finger tip to ride and feel this ridge. The ridge should be equidistant from the rim on BOTH sides. 
• Air Pressure. Modern tires need 10-12% MORE air pressure than the numbers listed in your owners manual. This will get you up to about 33-36 psi in the front. Less air pressure and the tire will wallow and not steer accurately.
• Fork Oil. Do yourself a favor and change the fork oil. 250cc of BelRay 7W or 10W in each leg please. No one ever changes fork oil, so the oil can get old or infused with water from pressure washing. Don't bother trying to "inspect" the oil, just change it ! You can buy a plastic baby bottle for 69 cents to help you measure. You may need a funnel. 
• Wheel Bearings. While facing the front wheel from the side, grab the tire at 2 positions opposite each other (e.g. 11 and 5 o'clock). Do a hard push-pull with your arms. You should feel ZERO "play" or looseness in the front wheel bearings. If they are loose, you may hear a slight tick-tick, or feel a small click-click. Wheel bearings go bad mostly due to pressure washing, but BMW also has a schedule for wheel bearing grease renewal.... which if you don't know when it was last done I highly suggest you do in the next 2 weeks. A tub of modern, black, synthetic automotive grease will do great. The job is easier than it sounds. You won't be removing anything but the outside seals and the rolling elements.
  ► If there is looseness, or if the bearings are dry, you MUST replace both bearings AND add new grease. A heat gun is needed.
• Balance. Suspend the front wheel by its own axle, which itself is supported at each end by new ball bearings with a 17mm ID. The wheel should stop at random positions. If it keeps stopping within 3" of the same position, then the wheel is not balanced correctly. Self-adhesive lead weights can be had at your LAPS. Static balance is all that's needed. [NOTE: You cannot balance the front wheel using it's own bearings due to the drag of the grease and grease seals.]
• Aging Tires. By law, there is a 4 digit year code molded into each tire on one of the sidewalls. Hopefully your tires have a xx15 (2015) or later. If this bike has been sitting, then it's not uncommon to see something like an "xx89" (1989) there and you MUST replace the front tire. Please don't trust tires over 5 years in age !!

  

Head Post Bearings. My last 4 Airheads each had congealed head post bearing grease. This "grease" had turned to a very thick, almost plastic feeling, putty. The OEM greases pre-date synthetic grease and will therefore oxidize and thicken. Due to the gyroscopic effect of the spinning wheel, ALL front ends make quick, self-induced micro-adjustments as the bike navigates the road surface. When the bearing grease is this thick, then instantaneous corrections are not possible... the slower reaction times then can result in "wobble".
Replacing the grease is actually a very simple job. Remove the bars and fork crown. Loosen the head post nuts and allow the front end (complete) to drop 3-4" to a supported position. The top bearing can be removed; the bottom bearing can be washed in situ. The bearings are rarely "bad". Wash out/ blow out the old grease. Add new grease. A thick, synthetic, water-resistant grease is advised. Unlike most motorcycles, the Airhead has a specified head post bearing torque to enable bearing "pre-load". 

• 
  Access to the top bearing

• 
• Access to the bottom bearing.

All these repairs, done as a group, will fix 99% of all front end issues, and you'll be able to ride with confidence knowing the issue will not suddenly re-appear at 60 MPH.

Hope this helps.

@wobbly That's a wealth of information!  Thank you.  I've had the spokes checked and the tires re-balanced.

The wobble is still apparent (at speeds around 30-40 mph) although intermittent.  I'll tackle the fork oil idea next.

Cheers!

@4949 Thanks for the tip!  I've had the spokes & wheel balance checked.

Next it's the fork oil and head bearings...

@wobbly You got it!  Tires were 11 years old.  I put on Bridgestone Battleaxe tires and the wobble is gone.  Front tire went from 4 weights to needing no weights!

Glad it worked out.