They look great! I've been a carpenter in the PNW for decades, and would like to advise you that 2" screws attaching 2x4 to 4x4 is weak. I'd recommend adding 4" x 5/16" lag bolts to each place where your rails attach to your posts. At least use 3.5" deck screws if they begin to become loose after moisture affects them.

In order to avoid that last bit, moisture penetrating the joints and affecting the fasteners, I'd get two or three tubes of really good caulk, like ISO Quad (It's very waterproof, finishes smooth, and takes paint well) or something similar, and carefully seal the top and sides of every joint with posts. Make sure to leave no gaps at all where you caulk. I'd leave the bottom of those joints open so that moisture that gets into the joint can drain out. Gravity is your friend on the way out, but your enemy on the way in. I'd also make sure to completely seal with caulk every fastener, and particularly the lag bolts and cracks in the wood, so that no water can intrude and soften the wood in the hole. Using 2" screws to fasten 2x4 lumber only provides a half inch of penetration into the posts, and it doesn't take much to loosen such short screws. Make sure to inspect and recaulk yearly, as cracks in the caulk or paint can allow water in and it's a lot easier to take a few minutes and recaulk where needed than to replace posts or rails if you don't maintain them. Capping your posts also really prolongs their life.

Keep a close eye on the seal between the concrete and the posts, as even a tiny gap will let water in, and if water is able to stand between the concrete and the posts (in the hole in the concrete), they can rot very quickly. This is why I prefer gravel to set posts in, as it is just as strong in compression as concrete, but does not let water stand against the posts and drains well. Also, making the holes a foot deeper than necessary and filling that space with gravel helps a lot. If there seems to be gaps, even very small ones, you might caulk there too, in the hopes of keeping most of the water out.

I assume you're aware that your old fence boards probably won't last very long in constant contact with moist soil, so also commend you on recycling used lumber for such a short term job. The paint will help a lot, but you won't be able to maintain that paint without digging up your garden, so you can expect to replace them in a few years, especially since the fence was already in poor condition. I have lined the interior of raised beds with plastic sheeting, but that doesn't help much because where the plastic is in contact with the boards, it keeps a film of water there, and I suspect that moist dirt might actually be preferable =p

I've seen folks use hardware cloth, a wire mesh of 1/4" made of steel and galvanized, to underlie their raised beds in order to keep out moles and gophers. If it becomes a problem in the future, it's something you can consider. Gravel doesn't really stop them, and they can tear up a garden bed from the inside out.

May you get many years of enjoyment from your beautiful garden beds.

Thanks!