I know that this is a difficult issue that many of us struggle with. By the time a typical student reaches college, they have had over a decade of math classes and teachers. As such, most of your students will have likely decided whether or not they're a "math person" by the time they're sitting in your classroom. You have to confront that an individual might be capable of calculation and dislike math, be poor at computation and appreciate math, or be poor at computation and dislike math (I would argue that there is a correlation). Regardless of the specifics, you're fighting against phobias.
Try to maintain a level of empathy for your students---as a lecturer in math, I would wager that math was always an easy/easier subject for you throughout school. Even if you didn't necessarily fall in love with math until you took a class from Prof. X or read a book by [famous mathematician], you've likely always been good at pushing symbols around and solving for $x$. Don't take these skills for granted in your students. Try not to combine algebraic steps on the board, give examples like $\sqrt{3^2 + 4^2} \overset{?}{=} \sqrt{3^2} + \sqrt{4^2}$, try to write questions that don't hinge on a clever simplification step. Furthermore, encouraging students to work and study in groups can be supremely beneficial: hopefully they'll be able to correct each other's mistakes.
As for the motivation question, why do you think that undergraduates studying liberal arts should learn calculus? Personally, I would argue that the discovery of calculus is perhaps the greatest intellectual achievement of humanity over the last millennium. What do you want your students to know about calculus at the end of the course? Once they graduate? In ten years? Try to strike a balance between needing to learn rote computation, motivation, and learning the sorts of problems that mathematics can solve. For example, having your students learn enough mathematics to recognize that there might be a mathematics question lurking somewhere and hint hint hire a mathematician to help them out can be a useful outcome. Try to keep them from unnecessarily reinventing the trapezoid rule.
The history of calculus is rife with stories and anecdotes that can be incredibly compelling to students of history, literature, anthropology, and other liberal arts. Consider, e.g., the historical rivalry between Newton and Leibniz, the dysfunctional Bernoulli family, the tragic lives of Abel & Galois, the autodidactism of Ramanujan, the prejudice that Sophie Germain (Monsieur LeBlanc) had to fight. You don't need to provide an entire history of mathematics for every topic, but even just an offhand mention can help engage students: "Did you know that a mathematician named Evariste Galois proved in the 1800s that there isn't a thing like the quadratic equation [point to where you just used this on the board] doesn't exist for polynomials of degree five or higher? The legend goes like this: after challenging another man to a duel, he realized that he was probably going to die in the morning (again, he was a mathematician) and stayed up all night producing about ten pages of mathematics that graduate students still study to this day. He died at the age of twenty." Give them a reason to see calculus as a human endeavor and not just an edifice of facts. I have even gone as far as having calculus students write essays about historical figures.
As you mentioned with your analogy of using salt in cooking, try to motivate definitions and theorems with everyday experiences. It's difficult to imagine our modern understanding of the universe without the influence of calculus. Mean Value Theorem? Average speed over a road trip. Linearization? What was the weather like when you walked into the building...predict if it's nicer out now. Exponential functions? Account balance under interest. Derivative? How can you figure out your speed if the speedometer is broken. Integration? How can you figure out how far you've driven if the odometer is broken. One tip that I've found is to explicitly not use introductory examples from physics. While we might find the notion of a swinging pendulum or accelerating rocket intuitive, it is surprising how many students struggle with these analogies. Even the example of throwing a ball under the effect of gravity trips people up (although you should absolutely address this misconception!). Most students have poor intuition when it comes to science, e.g., this article from The Atlantic or this article from the Harvard Gazette or this list of Physics misconceptions by New York Science Teacher or this list of misconceptions on Wikipedia. Being able to correct even a few of these is also a good objective for a calculus course. I personally find that economics/money or driving a car to be the most accessible areas to pull motivating examples from, before addressing physics examples with your newfound machinery.
As a parting thought, try pondering the flipped situation: you presumably had to take some liberal arts courses as an undergraduate for your presumably math-or-related degree. What did you get out Foreign Language / Art / Music / Literature / Psychology / Economics 101? What do you remember? Why was it important that you take those courses? Did you have any particularly effective instructors?
