I'm not exactly sure what's in that seed staring mix, but I'd be willing to take a guess. It's probably peat moss, perlite, and some fertilizer.
Peat moss is great at water retention, which is why many people use it in seed starting mixes. The problem is that it becomes extremely hydrophobic once it's dried out and does exactly like you describe. It draws up tight and the water runs over the surface, down the sides and out the bottom, letting no water get inside the soil.
I've been planning to experiment with compost, bone and blood, meal and worm castings I've been making, but what I've done for years is use regular miracle grow potting soil and add perlite to it.
The reason you add the perlite is to break it up and make it really loose so the roots can easily penetrate the soil. You don't want them to have to spend energy fighting to push out their roots.
There are a few reasons why your particular seeds may not be germinating. I've never actually tried to grow rosemary and dill from seed, but these will generally apply to your situation.
The first issue is that you have to think about how these seeds would grow in the wild/your garden if you let them alone. The plant would run to seed, the pods would open, the wind would scatter them nearby and they'd sit there all winter till the spring. What happens in the spring? The ground starts warming up. Many plants are programed to need warm soil before they'll germinate. It varies depending on the plant, but I'm going to say around 80+ degrees as a general rule. They sell heat mats you can set your seed starting trays on and thermostats that control the heat. These are pretty cheap.
So you have to research your particular plant, because some plants actually require that cold period, because they don't want the babies to come out till spring when they can survive. If they let the seeds go in the autumn and there is a warm day and they germinate, then they die in the winter. Other plants don't care and if they conditions are right, the germinate anyway. There are even plants that require you to physically damage their shells, like a lot of oak acorns, and then go through the cold temps of winter. I'm totally guessing, but I would think it has something to do with the damaged seeds being the ones that are probably carried off by squirrels or some other animal that gnaws on them some. This would insure the nuts have some chance of being carried away from the tree before they sprout so it's not competing with the parent. I'm sure someone will correct me if there is a legit explanation.
Then you have depth of planting. Some seeds require that they be planted .25" deep. Others just want a light dusting on top, and some need to be directly on the soil surface and won't germinate if they don't have sunlight on them.
The most common reason you can't get a packet of seeds to germinate, by far, is you might just have bad seeds. Seeds have a shelf life. Companies that sell them know this and they try to sell the old ones before the new ones. Also, the seeds might have left the seed company as brand new viable seeds, but they might sit around on the shelf at the store where you bought them for a long time. I doubt they'd throw money down the toilet because they bought the seeds last year, or even two years ago. This isn't necessarily them trying to sell you bad seeds or not caring about your. It's possible for old seeds to germinate. As a matter of fact, you should go look for a youtube channel called MIGardener. They are putting out update videos on a project where they found a cultivar of tomato that's supposed to be extinct that's been in a shadow box for years and years. They opened it and discovered the packed was dated 87 years ago. They tried to germinate all the seeds in four different packs of vegetables and the only viable plant was a single tomato plant. It's doing well, currently, and you can search 87 year old tomato on their channel or youtube in general and see it. So you may just have bad seeds.
So do a little more research on the requirements of these plants. I don't know about starting these from personal experience, so I hope someone else can answer that part for you. Again, it might not be you and might just be bad seed.
The last thing I'd recommend, especially if there is peat moss in your starting mix, is don't water from the top. You're dealing with delicate plant and top watering can bury the seeds too deep, wash them away, or damage these microscopic babies. I'd advise bottom watering till they get a little size on them. Just set whatever container you have them in a proportionately shallow tray of water. Just keep the tray of water topped off. The pot will soak up just what it needs. The soil in the bottom of the pot, level with the water in the tray, may become water logged, but it's not a problem. if the roots don't like it, they just won't grow into it.
This isn't a viable long term solution to watering your plants and the reason is because of salt build ups. Salt is actually necessary for plants, in small quantities, but especially if you're fertilizing the plant, you'll be adding small amounts of salt or their may be salt in your water. It builds up and it's being removed, because your pot is just soaking water up from the bottom. In nature or even in regular pots, you often top water, which flushes salts and other buildups out the bottom of the pot. If you always water from the bottom, the buildups can kill the plant.
I wouldn't worry about it with your setup, because you're just starting seeds. By the time there is any worry about build up, you'll have repotted them long ago. I also don't see it being a huge issue with vegetables and herbs anyway, because they're usually seasonal and I'm not sure how much salts build up in one growing season. Even so, it's a viable watering solution for some plants. I have carnivorous plants that I have bottom watered for years. The kicker is that they're outside all the time and every rain storm flushes the pots. If it hasn't rained in a long time, I'll sometimes flush from the top, but I rarely have.
Give all this a try and see what happens. You might also want to just try a new packet of these seeds. My recommendation for you as a beginner is that you just go to your local big box store and buy these plants. They aren't going to cost but a few dollars and you're going to get more satisfaction out of having the plant and being able to use the herbs than you'll get being frustrated trying to grow these seeds. I think you should still keep trying the seeds, because it's good/fun to learn and expand your skill, but I'd have the seeds as a backup and just buy the plant for now. I've been doing plant stuff for years and I still experiment. This year I'm going to try air layering stuff for the first time.