Short answer: both forms are possible in both sentences.

I searched the web a little for inspiration and found a lot of sites with incorrect information, so beware. Some sites say that it’s always “à qui” for a person, which is wrong. Some sites allow both but then state that certain example sentences can only use one or the other even though both are possible (but may have slightly different meanings).

Taking personne as an example, “personne à qui” is slightly more common than “personne à laquelle” and the ratio has been pretty steady for the past two centuries.

There is a general rule that the relative pronoun is:

• qui as a subject;
• que as a direct complement (COD or attribut du sujet);
• lequel/laquelle/lesquels/lesquelles after a preposition other than à or de;
• when the pronoun comes after the preposition à, the two words must be combined into auquel/à laquelle/auxquels/auxquelles;
• when the pronoun comes after the preposition de, the two words must be combined into duquel/de laquelle/desquels/desquelles;
• for people, qui can also be used after a preposition: à qui, de qui, pour qui, etc.
• both de qui and duquel/de laquelle/desquels/desquelles may or must sometimes be replaced by dont, and it’s complicated;
• quoi only after a preposition when the pronoun refers to an unspecified thing and there is no true antecedent: “ce à quoi je pense”, “ce pour quoi je me bats”;
• où for a time or place.

There may be more cases that I’m not thinking of. Note that the words qui, que, quoi, où are also interrogative pronouns, but chosen in completely different ways.

When do you use à qui and de qui for people? In many cases, both are possible. It seems that both natives and non-natives sometimes attempt to draw firm rules, but usage does not confirm these rules.

Many people state that one of à qui and auquel/… is more refined than the other, but… they disagree as to which way it goes! Many consider auquel more refined, perhaps because lequel as a subject used to be widely used, but is now significantly stilted and mostly found in legal texts. However, it seems that Girodet disagrees:

Lequel employé après une préposition. —
L’antécédent est un nom de personne.
Lequel (auquel, duquel, etc.) peut toujours être remplacé par qui. Ce remplacement est même conseillé dans la langue élégante : Ces camarades pour lesquels (ou mieux pour qui) j’éprouvais une sympathie assez vive.

Littré, who is usually very opinionated, does not take a stance either about lequel or qui.

I can only find one thing that does make a difference between lequel/duquel/auquel/… rather than ∅/du/à/… qui. We tend to use lequel/… when the antecedent is a person or group of persons having a specific property. In other cases, I think there’s a free variation (i.e. the same speaker might use one or the other pretty much indifferently), but with qui being more common. In the following examples, both forms are possible, but I only wrote the form that I’d favor.

Les gens à qui elle a parlé ne lui ont pas répondu.   (The people she spoke to did not answer her.)
Les gens auxquels elle a parlé étaient les plus susceptibles de répondre.   (The people she spoke to were the most likely to answer.)

Thierry est l’homme à qui je pense sans arrêt. Il m’obsède.   (Thierry is the man I can’t stop thinking about. I’m obsessed by him.)
Thierry est l’homme auquel je pense. Aucun de ses collègues n’aurait fait cela.   (Thierry is the man I’m thinking of. None of his colleagues would have done this.)

La personne à qui j’ai indiqué le chemin portait une robe verte.   (The person that I pointed the way to wore a green dress.)
*La personne à laquelle j’ai indiqué le chemin est arrivée la première.   (The person that I pointed the way to arrived first.)

This distinction is somewhat similar, but not as sharp, as the distinction between qui and lequel/… as an interrogative pronoun, which is very well explained in an answer to a related question.

Correct: L’homme à qui je pense

Correct: L’homme auquel je pense.

I’ve found the source for this, it’s here, but it’s wrong.
Be careful with sites explaining grammar on the Internet.

They give as examples:

Thierry est l’homme auquel à qui je pense.
La personne à qui à laquelle j’ai indiqué le chemin.

So, why they consider “homme” with a different grammatical role than “personne”.
It makes non sense.

And they add:

On emploie auquel et non à qui pour désigner un objet ou un animal.

So “personne” is an object of an animal??

There’s no rule like: you can’t use à laquelle, when you talk about a person.
This rule doesn’t exist.

C’est la personne à laquelle je pense.

Is perfectly correct.

C’est l’homme auquel je pense.

Is perfectly correct.

In short, “auquel” and “à qui” are equivalent, but “auquel” is more formal, and seems often more correct to me (some speakers prefer the first one, some others prefer the second one )

See here: https://la-conjugaison.nouvelobs.com/fle/les-pronoms-relatifs-composes-79.php

Pronoms relatifs contractés- auquel, auxquels, auxquelles
Ils sont issus de la contraction de la préposition à + lequel/lesquels/lequelles ( pronoms relatifs composés)

Exemple :
Les documents auxquels (à + lesquels) je pense sont dans la salle des archives.

C’est le garçon à qui (ou auquel) elle pense depuis un an