I understand her situation is tough but she also doesn't want to help herself
Way to contradict yourself within a single sentence.
and just passive aggressively wants to drag Sakura down too.
Near as I can tell Ashima's feelings about Sakura are one big conflicting mess. On one hand, she's one of all of like two people in the entire town, and only one in the family, who Ashima can think of as "being on her side" even if she really can't do very much to concretely help (Mayumi doesn't exactly count here since Ashima A) has basically lost all faith in teachers and the like to begin with B) currently perceives her as an intruder violating her one real refuge at Rin's). It's not like she doesn't appreciate the sentiment, and she does seem to return at least some of Sakura's Mochi Seal of Approval beyond-sisterly affection too.
OTOH, Sakura has the social and familial position that has been arbitrarily denied Ashima through no fault of hers and this is something their surroundings rub her face in daily at ever turn. Ashima's need to be the goddamn Buddha to not feel hella bitter and jealous about all this, and tempted to vent her misery at Sakura - who is pretty much the only one she can take her frustrations out on anyway - on occasion.
All of which practically appears to boil down to Ashima's treatment of her sister fluctuating wildly depending on her current emotional state. If she's been served a particularly stinging rebuke of the colossal unfairness of her situation, as with the shoe incident back when Sakura was first introduced, she's wont to pass it along by being passive-aggressively mean. If she's in a stabler mood (presumably having been left in relative peace recently), as in the most recent chapter, she's pretty friendly in her particular taciturn fashion albeit still prone to yanking Sakura's chain so to speak.
Sakura, for her part, seems determined to grin and bear it all. Partly because she's gay for her sis obviously, partly, I suspect, because one of the very few concrete things she can do to make Ashima's legitimately shitty situation a bit more bearable is to accept some of the burden by proxy when the latter needs to unload. Plus she likely has one Hell of a nagging conscience about the whole situation and her own inability to do diddly squat about it, as well as her own in comparison quite privileged and pampered position. A way to make amends and even the scales even a little bit, so to speak.
All of which isn't really very healthy behaviour from either of them, of course, but then what do you expect in the context? They're two middle-school kids stuck in a situation that's practically psychological horror - "Hell is other people" as it were.