Great to hear that things are proceeding with Amber Ellen.
As Trevor says ‘Paper F Keel Servicing’ on the Technical Papers page is an essential start point. It gives a really clear explanation of the various components, their operation and repair. (Its worth familiarising yourself with the other keel related papers too.)
You ask about support when out of the water. It is certainly a good idea to lower them down onto the trailer structure or onto blocks on the ground to relieve the strain on the gear. Equally it helps to loosen all three keel bolts on each side and try to lubricate the mechanism and particularly the clamping plates when laying up. There should be a way of removing the keel box ‘lids’ forward of the bulkhead to gain (limited but sufficient) access for this lubrication.
As the keel are stuck getting some penetrating oil all over the clamping plates now, perhaps repeatedly over a few days, would be a good thing to do regardless (with due regard for any woodwork you might be going to paint).
Have you managed to move all six keel bolts? By which I mean slacken the nuts off but do not remove so they are a few millimetres (should that be in inches for an Atalanta?) off their tight position and then hit them to move the keel bolts towards the outside of the boat. Before doing this you should remove the keel bolt covers on the outboard sides of the keel boxes. There are many different arrangements but typically each side will have three disks held to the case by bolts or screws.
Can you post a photo of this area of the keels? It is surprising how much weight a little corrosion in the plates can support. When trying to remove the keels on Emma Duck last year we removed all the keel bolts and went at the keels with a scaffold pole and sledge and they were still reluctant to come out! (I am not suggesting you follow our method!)