Pazara gidince taze kayısı almadan edemiyorum.

Here it means market.

Turkish pazar can mean either market/bazaar or Sunday, but the context makes it clear:

• pazara gitmek = to go to the market
• the sentence also talks about buying fresh apricots, which fits market naturally

In fact, pazar often suggests an open-air street market, especially for fruit and vegetables.

The -a is the dative case, which often marks direction: to, toward.

With gitmek (to go), the destination is usually put in the dative:

• eve gitmek = to go home
• okula gitmek = to go to school
• pazara gitmek = to go to the market

So pazara means to the market.

Gidince means when (someone) goes or whenever (someone) goes.

It comes from the verb gitmek (to go) plus the suffix -ince, which often gives a meaning like:

• when
• once
• whenever, depending on context

In this sentence, because the main verb expresses a repeated tendency, gidince is best understood as whenever I go or when I go.

This is because gitmek is one of those verbs whose stem changes in some forms.

The dictionary form is gitmek, but before vowel-initial endings, the stem often appears as gid-:

• gidiyor = he/she is going
• gidecek = he/she will go
• gider = he/she goes
• gidince = when he/she/I go(es)

So gidince is the expected form here.

Yes. Almak can mean both to take and to buy/get, depending on context.

In shopping or market contexts, almak very often means to buy:

• ekmek almak = to buy bread
• meyve almak = to buy fruit
• kayısı almak = to buy apricots

So here almak is naturally understood as to buy.

In Turkish, nouns are often left in the singular when speaking about things in a general or indefinite way, especially food or items being bought.

So taze kayısı almak naturally means to buy fresh apricots.

Using the plural kayısılar would sound more specific, like you are talking about particular apricots already known in the conversation. The singular is the normal choice here.

This is a very common Turkish pattern:

-madan / -meden edememek

It means:

• to be unable to refrain from doing
• to be unable to resist doing
• to be unable to help doing

So almadan edemiyorum means:

• I can’t help buying
• I can’t resist buying
• I can’t go without buying

A rough literal breakdown is:

• almadan = without buying
• edemiyorum = I can’t do/manage

Together, the idea is: I can’t manage to leave without buying some.

They are not a simple double negative. They belong to a fixed structure.

• almadan = without buying / not buying
• edemiyorum = I can’t manage / I’m unable

So the meaning is not I don’t not buy.
It is more like:

I can’t manage not to buy.

That is why the whole expression means I can’t help buying.

Because Turkish often leaves out subject pronouns when they are already clear from the verb ending.

In edemiyorum, the ending -um shows I.

So ben is unnecessary unless you want emphasis:

• Ben pazara gidince... = As for me / I, when I go to the market...

Without ben, the sentence sounds completely normal and natural.

Usually it sounds like a general habit or repeated tendency:

Whenever I go to the market, I can’t help buying fresh apricots.

That reading comes from:

• gidince = when/whenever I go
• edemiyorum in the present = a current, repeated tendency

In the right context, it could refer to a present situation, but the most natural reading is habitual.