HI! I know your problem. I have the same bike.
ALL ORIGINAL AGILITY bikes do the same and need to warm up the engine before start riding below approx. +10 celsius. Because until the engine is not warm enough: if you pull the throttle, maybe the engine will stall. I know that it can be very exasperating. This warm-up period takes about 1-10 minutes depending on the temperature. E.g. in summer it is not necessary...
Why is this problem? In winter conditions the air is too much and the gasoline is not enough. I think the engineers wanted to minimize the fuel consumption&emission, and they thought that most people use their bike over +10 celsius - That's absolutely normal. Over this temperature there is no problem.
What you can do?
There are alternative solutions e.g. covering the air-cooling fan for some minutes in winter but that's not the best, especially, if you forget to remove the covering and the engine will be too hot.
So I recommend you to replace the carb's main jet for a bigger one (try to test how bigger is the best) and use with it at least in winter. I think in summer this bigger main jet is not necessary because of the higher fuel consumption, but I haven't tried it yet, because I'm a hypermiler. Maybe, if I'll use my bike next winter I'll also try a bigger main jet, but only for winter-period.
But, if you haven't had such problems so far, you must have set inproperly the carburetor. Unfortunately I can't help you to repair it in writing, you should try to reset it again properly. If you should not manage to set it, you may ask for an expert's assistance.
Greetings from Hungary
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