Full credit to @AxiomaticSystem for this answer (2nd problem). I'm just writing my personal logical deductions. Apologies for how long this is; I can make deductions but struggle to explain them succintly.
First off, let's figure out the vertical line. If it goes down, then we must consider how to fill the space next to it (2 to the left of the n). The only options are the t and the n. If it's the t, then it is part of the curvy bottom bit. The t's trunk is forced into that column, and with the curve and horizontal line there isn't any room for the n (picture to prove it:)
If we attempt to force the n into that spot, then there are empty spaces in between the two vertical lines, so that is impossible. (around this point I realized that I couldn't take pictures of all of the steps. Also, the timer on the web version was freaking me out, so the rest of the pictures are drawings on a screen shot)
So the vertical line must go up only. There are also some other lines forced. The t's letter must be on its right tip. Proof: It can't be on the top tip (can't go down), left tip (can't go right), bottom curve (can't go far enough up). The n's right line must go all the way up, since no other letter can reach the spot below the horizontal line. Then it has to go horizontal for at least 1, forcing the horizontal line to also go horizontal. So now the board looks like this:
Now turn your attention to the third column from right. It can be filled by the swirl, the t's trunk, the t's bottom curve, or the n. It can't be the n, that would leave empty spaces. It can't be the t's trunk, that leaves no space for the n. It can't be the t's bottom curve because then the four top-row spaces above the t cannot be filled. If one's filled by the bottom curve, then only one more can be filled by the t's top tip. If one is filled by the tip, no letter can reach to the space on the left of it. If none are filled by the tip... that doesn't work, there needs to be a tip. So it's impossible. If that explanation made no sense, look at the picture and try to fill the spots above the t.
So that column (3rd from right, remember?) has to be filled by the spiral swirl thing. To reach it, the spiral must curve around the edges. (has to avoid the h, t, and vertical line)
Now look at the h. Its long side has to be at least 3 long, so it extends into the available space. Then its shorter leg must be in the column next to its long side, or there would be unreachable space in between the legs (like there was with the n). So the h must look like this:
Now it's fairly trivial to figure out where the t's curve goes, which forces its trunk into the 4th-from-right column. Once the t is in place, the spiral just takes up the rest of the space
That was long. But there you have it: logical deductions for problem #2.