Learn Turkish: Bu problemi aşmak için bir psikoloğa gitmek istiyorum. - I want to go to a psychologist to overcome this problem.

It is the specific accusative suffix. Because you are pointing out a specific problem with the word bu (this), the direct object of the verb aşmak (to overcome) must take the accusative case, making it bu problemi.

What does the combination of aşmak and için do here?

When you pair a full infinitive verb (ending in -mak or -mek) with the word için, it translates to 'in order to'. Therefore, aşmak için means 'in order to overcome'.

Why does the word psikolog change to psikoloğa?

Because you are going 'to' the psychologist, you need the dative suffix -a. In Turkish, when a word ends in g (which often happens in foreign loanwords like psikolog), adding a vowel suffix causes that g to soften into a ğ (soft g), resulting in psikoloğa.

How could we apply the -dIkçA suffix from this lesson to these words?

This original sentence uses için to show a single direct purpose. If you wanted to use -dIkçA to show an ongoing or proportional action, you would attach it to the verb. For example, Bir psikoloğa gittikçe bu problemi aşıyorum means 'The more I go to a psychologist, the more I overcome this problem'.