Based on my experience, the answer is in your question:

Is this a case where some meanings are just more used than others, but you would have to be immersed in the culture to understand which is more often used?

In my Country (Cameroon), French and English are national languages (Here we use to say “Cameroon is bilingual not cameroonians 😀 ). We also learn German and Spanish at school as LV 2 = Langue Vivante 2.

I will underline Vivante which means you need to live a language to learn it well.

The two needs of a language are Grammar and Vocabulary and, I don’t believe you will be able to get your Vocabulary from books.

— Est-ce que tu veux de la glace ? — Oui, j’aime assez la glace

To a native French, this is clearly enough context:

• “Would you like some ice cream?” “Yes, I like ice cream quite a bit”

« Désolé, je suis toujours malade »

Well, this is a bit more ambiguous, but depending on who is saying it, one should be able to quickly decide on the most likely meaning of it, extreme cases being some athletic young person, strongly built but looking tired or greenish (then we’d lean towards still sick), or some weak old person struggling to make more than a few steps (then quite possibly in a semi-permanent state of weakness).

Also and importantly, as Améraldor rightfully pointed out in the comments, the way toujours is pronounced would usually be a perfect clue as to how to interpret it:

• Désolé, je suis TOU-jours malade ! _{(TOU- louder and/or higher pitched)}
  Désolé, je suis TOU-JOURS malade ! _{(TOU-JOURS louder and/or higher pitched)}
  Désolé, je suis touuujouuurs malade ! _{(toujours longer, sometimes also higher pitched)}
  → Sorry, I am AL-ways sick!
• Désolé, je suis toujours malade !
  → Sorry, I am still sick!

There has been comments made about a statistical way of finding the meaning of the sentence, claming that the version “always sick” would usually be worded “tout le temps malade”, so therefore “toujours malade” should be “still sick”.

I am unsure what dataset was used to make these claims, and they may be valid, but as far as I am concerned, I would use “toujours” most of the time. There are therefore limitations to this method, and the tone method pointed out by Améraldor is a lot more reliable in my opinion, since it would work even for particular individual whose choice of words might be off the claimed standard.

« Je rentrais du tennis. Comme d’habitude, après une bonne heure intense, je suis assez chaud et j’ai envie de baiser mon copain. »

Well, after an hour of tennis, and intense tennis that is, the need for a two-second interaction through kissing a boyfriend, or more generally a lover, would be a weird one to express. Though the longer type of interaction, the one unambiguously meant here, is also somewhat weird to express to a third-party.

_{And as mentioned in the comments below, the verb baiser is nowadays used mostly, if not exclusively, to indicate a sexual intercourse. It is perhaps the only acception used in Europe (I’m not sure, but the comment makes it sound like it), but in Quebec, the other acception (to give a kiss) is still used, though not as widely as before, and not in the majority of the cases anymore either. I was pointing in that rought direction when I used the word unambiguously, but it might have been a little too discrete to do the work. So here is the full statement. Thanks to Jeremy Grand.}