It wasn't until I re-read the book (although maybe that means I'm just slow at picking up on these things) that I realized Dalinar had the perfect way to remove any suspicion that he tried to get Elhokar killed during the chasmfiend hunt, namely that, during said chasmfiend hunt, Dalinar saved Elhokar's life! If Dalinar had wanted Elhokar dead, he could have just let the chasmfiend kill him; instead he leapt in front of the creature's giant claw and actually caught it between his arms, saving the king's life. Why neither Dalinar, Adolin, Elhokar, or Sadeas ever brought up this point is a mystery to me.
Elhokar's paranoid enough that he wouldn't take that as proof, and Dalinar, Adolin, and Sadeas all know him well enough to know that. Remember what Dalinar had to put him through at the end of the novel to convince him?
I believe the reason was that they thought it was a gambit by Dalinar to get Elhokar to trust him, though I can't remember if this was explicitly stated in the text or something that I thought up on my own.
Also keep in mind that it's not so much an issue of proving Dalinar wasn't involved. It's an issue of Sadeas using an accusation of Dalinar's supposed guilt as justification to go to move against him with the grinning approval of the other Highprinces.
Shallan's money trouble (contains major spoilers)
Shallan secretly posseses a shardblade that she apparently took from her father after she killed him. Shardblades are worth "Fortunes. Cities, palaces, kingdoms." If her family is in such terrible enough debt that she is desperate enough to try conning and stealing from the sister of the most powerful King in Roshar then why does she not just secretly sell the blade. It can't be harder to arrange than what she tries to do instead.
There's implications that there's something much worse going on in the background surrounding Shallan's family, even beyond the Ghostblood connection, and that trying to sell the Shardblade would simply result in things getting worse. Blades aren't things you just sell in Roshar, and merely possessing one will bring down tremendous and terrible attention on the family. Attempting to sell the Blade would likely end in the whole family being killed and the Blade taken. Trying to sell it is like trying to sell nuclear weaponry in the modern world.
Is trying to sell nuclear weaponry that much more dangerous than trying to steal nuclear weaponry, the soulcaster is probably just as dangerous and valuable as a shardblade. Both prospects are just as risky but in opposite directions, the soulcaster is already in the posession of a dangerous powerful person but once taken will be kept hidden, the shardblade is already hidden but for it to save them would require it to be brought to the attention of potentially powerful dangerous people.
Stealing the soulcaster is obviously the less risky proposition, as it would not draw attention if done properly. The shardblade would draw attention regardless.
Also, they have a soulcaster that she plans to replace the stolen one with. The idea, was that Jashna would think her soulcaster had broken, fix it or find another stone to replace the broken one, and there would be no harm no foul.Or, failing that, that Shallan would be gone by the next time Jashna tried to use it anyway. From my understanding, the reason they can not fix their own family's soulcaster is because none of them have the knowledge to fix it, and are afraid to look to outside help because it would make them appear weak and vulnerable (and also because, I believe, their family has not et others know that the reason they have been as profitable as they have been was because they had a soulcaster).
They were never supposed to have a soulcaster in the first place, so yes, they couldn't get outside help getting it repaired. The people who apparently gave it to their father are also the ones who they suspect will kill them if they find out it's broken.