Winterizing:
You do want to drain or fill with antifreeze. Rubber and flexible plastic hoses can (usually) freeze without harm, but rigid materials will crack. In the water, I would close below-water thru-hulls but keep above-water ones open. Do drain your raw water strainer or fill with antifreeze. I would suggest you do the following procedure, after the engine is fully winterized:
1) Open the intake thru-hull, open the strainer and let it drain to waterline level.
2) Close the intake thru-hull, fill to top with antifreeze, open thru-hull and let it drain to waterline level, then close it again.
3) repeat step #2. This leaves the entire intake line between strainer and thru-hull (including seacock valve) full of antifreeze.

Fill any other below-water thru-hulls with antifreeze to the best of your ability. Being in contact with water will help keep them from freezing, but there are limits to that.

Battery:
Most batteries with 1-2-BOTH combination switch are wired with the negative wires always joined together and connected. The switching is done on the positive side. Basically, the 1 and 2 positions are on/off state, and the BOTH positon connects both in an on state. Usually one is dedicated to start the engine and the other to the house bank. (The thinking is that you join BOTH when charging, or when one or the other is getting to low so the remaining one can be switched in as a reserve. It's not such a great system but that's a different discussion.)

If either of your batteries can start the engine when switched to 1 or 2, then they are always in a BOTH state and I suspect a prior owner has changed connections around (so it's up to you to figure it out, but we can probably help if you ask questions).

For trickle charging - they should normally be connected as BOTH. With a 5-10 watt 12v solar array you would normally not need a charge controller, assuming your batteries are Group 24 or larger.

Hope this helps.