I find it confusing since we don't have a verb to match conseguir precisely in this usage. We'd say, "It is my custom," or use an adverb, "I customarily (verb)." I can infer a meaning by thinking "I follow" in the sense of following a custom, but it seems like a usage one would rarely if ever see.
I think of conseguir as "manage to" (+infinitive=manage to do, +noun=manage to get), which is a little bit different in meaning from "having the ability to do"- more what I think of with poder. Perhaps PaulEnrique or one of the native Portuguese speakers here can tell us if that's correct.
You are correct about "can" vs "to be able". There is a difference between "conseguir and "poder".
Both paulenrique and cloudhorizon explain in this post.
Caminhar is to walk exclusively by foot.
It can be an exercise walk, it can be a trekking walk, and it can be a contemplation walk (the spanish example you gave).
Andar is just walk. Nothing more.
In certain cases, conjugated with other words and contexts, it can be ride or "go by something", like "andar de carro" (go by car), "andar de cavalo" (ride a horse), "andar de avião" (go by plane).
Um carro pode andar, mas não pode caminhar (because a car doesn't have legs).
I think that in this context it takes the meaning of "to achieve" something. So I think that it means something along the lines of "I cannot achieve more walking." or "I cannot manage to walk any further". It is just another way of saying " I can't walk any more." but it kinda adds an implicit "even though I am trying to".