Don't worry about the heat shields just yet - I ran without them for a couple of years and while it didn't like sitting in traffic for prolonged amounts of time it didn't just run silly rich all the time. A "short run" should give no trouble.
Worn spindles will cause untold trouble keeping the things balanced and the idle smooth, but won't cause it to run rich.
When you say "almost closed the carbs" do you mean winding the jets all the way up (screwed in)?
If so, I'd be looking at the float level, the float valve (needle jet) sealing and - MOST importantly - the sealing of the piston diaphragms to both the pistons and the carb bodies. It's *very* easy to get them just a tiny bit off-centre when screwing down the retaining ring and/or the vacuum chamber cover, and any slight leak will result in that carb running very rich (vacuum leak = low piston = excessive airspeed under piston = rich mixture).
Do the carb-to-intake-elbow gaskets leave the air intake slot above the main air path and the small holes off to the side free? Lack of air flow to the atmospheric area under the piston will cause similar issues.
Also, what fuel pump are you running? Could the pressure be too high, resulting in over-filled float chambers?
Do the chokes turn off all the way? May be worth pulling the carbs off the manifold and stripping the choke units down - if the metering disc (with one with multiple small holes and the slot in it) is loose on the shaft (the mounting hole should be D shaped, with a flat - if it's worn the flat will be V or ^ shaped) the choke will never turn off as the slack will prevent it being rotated all the way back.
Blocked air filters? If the engine's tired and breathing heavily it'll clog the small surface area of the Sport air filters surprisingly quickly. I converted my Strom airbox to take a Fiat Cinquecento air filter: £4 from Halfords, so available everywhere at almost any time.
Some people will try and tell you that Strombergs are curse word: unreliable, leaky and prone to going out of balance. I disagree: be methodical in your approach, make sure they're not knackered, and you'll be fine. Probably my last 30,000 miles in the Imp have been fuelled by Strombergs, the first 25,000 of which were done on a pair of carbs thrown together with other people's cast-offs; they needed new jets, metering needles and float valves to make them *right*, but they were still eminently usable, and other than the occasional re-balance and mixture tweak didn't need any repairs.
Finally, I strongly suggest you invest in a set of Viton O-rings for the main jets - 3 O-rings per carb in 3 sizes. They are available from Imp Club spares, and IME are far superior than Nitrile when exposed to modern fuel - the Nitrile seals I first used started to leak badly after a fortnight, the Viton examples lasted years with just a tiny weep.