Mebbe the carbs need cleaning? Did it sit for a while before you got it? The 85/86 VT1100s were setup very lean and stock jets mean you almost always need to use the choke to start and until it is warmed up. Sounds like yours might be a little plugged. Can you keep it running after it is warmed up by using the choke? That would mean it has partially plugged jets = carb teardown/cleaning/rebuild. When I first got Lazarus, he had sat for a while, so I did this:
1. Drain the carb float chambers.
2. Pour in some fresh gas, blip tghe kill swithc to pump fresh fuel up into the carbs and let it sit overnight.
3. Drain the carbs.
Repeat until the gas coming out of the drains is clear and clean. Any greenish/yellowish color indicates jet corrosion 'cuz they are made from brass.
Another trick is to get a can of carb cleaner spray and after draining the carbs, spray up through the drain - that can cleanup stuff in the float chambers a bit. Again, repeat until the liquid coming out of the drains is clear. If there are still problems, the full teardown/clean/rebuild is probably necessary. Once you do go the rebuild route, replace the float needle valve and seat, gaskets, O-rings and adjust the float level. It would be a good idea to upgrade the jets a bit while you have everything opened up.
Rebuild kits - scroll down to the 1985-1986 kit:
Carb Kit Capital - Motorcycle Carb Parts
Jet calculator and kits:
Honda Jet Kits :: Honda 1100 cc Jet Kits :: Honda VT1100C Shadow 1985-1986 - Jet kits for Carburetors
Obviously, you will need a manual with diagrams and procedures. One tip - if at all possible, do not disconnect the cabs from each other, doing so almost always means a resync. If you are lucky, they are not too badly out of sync as is. You will prolly want to do a resync after getting it running better anyway, but why make more work for yourself at the start