|Fig. 1: わちきやア|
"Thinking that you might not know me anymore, my heart was pounding (fig. 1). And since I hurriedly walked, I'm in pain," Yonehachi said, beating her chest (fig. 2).
|Fig. 2: そしてもふ|
The only grammatically sound explanation I was able to find is that 「め」 is the realis form of the speculatory auxiliary verb 「む」. However, I have no idea what function 「へ」 has here (it doesn't appear to be the directional particle 「へ」). The following, interrogative particle, 「か」, is usually preceded by either a noun or the attributive form （連体形） of a verb with an implied noun.
|Fig. 3: 胭がひっつく|
"My throat is sticking," she said, sitting down next to Tanjirou (fig. 3).The first kanji in fig. 3 is yet another good example of non-standard kanji usage. The two acceptable kanji for 「のど」 ("throat") listed in Koujien are 「喉」 and 「咽」. While the right radical of the first kanji in this sentence is clearly 「因」, the left radical doesn't seem to be 「口」. To find other kanji with the same reading, I used Tangorin, which yielded one other kanji with the 「因」 radical on the right - 「胭」. I'm not sure if that's right, but it fits a lot better than 「咽」.
"Have you been ill?" Yonehachi said, intently looking at his face (fig. 4).
|Fig. 4: おまはん|
The next part of the sentence, 「煩つてゐさつしやる」, would be written using modern kana orthography as 「煩っていさっしゃる」. Unless you've encountered this particular pattern before, you have no choice but to check each permutation of small/large kana for the 2 instances of 「つ」 and 1 instance of 「や」 to try to find a matching word. In any case, the verb 「ゐさつしやる」 is equivalent to the modern Japanese verb 「なさる」, the polite form of "to do".
Also note the lack of iteration marks for 「つくづく」 - there are cases where repeated sounds/patterns are written out. Edit: As Matt suggests below, this may be because the repeated kana occurred at the start of a "new" line (at least within the togaki).
It's not clear whether the second to last character in the togaki (stage instructions written in two parallel lines) at the end of fig. 4 is the kanji 「見」 or the hiragana 「み」. Either one would make sense grammatically, and 「見」 is the kanji which this kuzushiji form of the kana 「み」 came from.