Reading ブラックジャックによろしく manga exercises (p14-and up)

Claiming Q: Part 1

当直『の』時だって自分『の』失敗ばかりこだわって

I’ve subdivided Q into two, ① and ②:

Picture subdivided

I’ll deal with them in two different replies.

Transcription ①:

With readings:
《当直【とうちょく】『の』時【とき】だって自分【じぶん】『の』失敗【しっぱい】ばかりこだわって》

Without readings:
《当直『の』時だって自分『の』失敗ばかりこだわって》

問題①:

You guessed it, my question is about the 〜て form here as there are two of them.

The first one is だって which I suppose is the sequential 〜て. However, I googled it and I found various answers. 当直の時 means the time for the hospital job and I think だって is the casual 〜て form of the copula です. So it could mean:
its the time for the hospital job and

Another interpretation for だって from Japanese stack exchange:

Several answers above attempt to make the distinction between だって as a contraction of the copula だ and the quotation particle と, and だって as meaning “because” or “even” or “but”.However, ALL of the various uses of だって can be traced to a simple contraction of the copula だ and the quotation particle と.

So which one is it?

ⓐ sequential 〜て form of copula です in casual form
or
ⓑ だ+quotation particle と?

The rest of the sentence:
自分の失敗: my own failure…

The rest I wasn’t familiar with. I pasted 失敗ばかりこだわって in my dictionary to see how it would end up being parsed:
『ばかり』is actually a particle meaning: merely, only, nothing but…
こだわる is a verb, “to fuss over”;

Is こだわって also a sequential here?

Edit: I’ve added U-Y at the beginning of the thread.