Congratulations on your baby and good for you for seeking the help you need to give your baby the superior infant food. Clearly, you are a proactive mom. I'm sorry you are sore and frustrated.
I don't know of an LC in Salt Lake (I'm in Ogden) but I have a lot of respect for Becky McInnis, a certified nurse midwife who practices at Birth and Woman Care at the Birth and Family Place in Holladay. You could Google her name or go to www.birthandfamilyplace.com to find her. She has more than 20 years experience of midwifery and I'm sure she would be able to help you. She also works with a couple of certified lactation consultants who teach breastfeeding classes and if she cannot see you, she could surely refer you to one of them.
You can also Google IBCLC, as that's the highest level of certification for lactation consultants, and you'd likely find someone who works through a hospital, if that's what you're looking for.
La Leche League is an awesome resource, and it's OK to call a different Leader and see if you get more information. I think you're smart to attend a meeting. In-person help, especially with a latch question, is more effective than phone help.
Clicking can be a sign of being "tongue-tied," which means your baby may have a short frenulum. Notice if her tongue curls up instead of sticks out past her gum line when she cries, or if it seems heart-shaped. Some babies really do have a short frenulum that needs to be clipped (a minor, in-office procedure) to nurse comfortably, but most often the clicking can be fixed with improved latch. I have found the graphic at www.kellymom.com to be useful--go to the "free handouts" section and find the handout called "when latching." It provides step-by-step instructions and a great illustration of the key points of a comfortable latch. This can help you evaluate what's going on and work to fix it.
Milk production is all supply and demand. Don't worry about causing chronic engorgement by expressing milk just to comfort; it's important to treat any engorgement and it will be easier for Baby to latch to a soft breast. Expressing just a bit of milk before latching can also remove a bit of foremilk so Baby doesn't get such a big initial gush, and receives the creamy, high-fat hindmilk sooner. You can find hand expression techniques (the Marmet technique is safest and most effective) at www.llli.org.
Consider nursing against gravity, such as tipped back in a recliner. Consider nursing while lying down, so any excess milk can run out onto a blanket and Baby can re-latch without gagging on a big let-down. Consider nursing on only one side at each feeding to ensure Baby gets hindmilk and to help regulate your letdown reflex a bit on the unstimulated breast. Consider nursing very often, according to your baby's cues. Consider relatching as needed, rather than putting up with a not-so-great latch. Consider treating any nipple trauma with Lansinoh lanolin to prevent any scabs from forming.
Adding artificial baby milk and/or a bottle into the mix at this stage will just complicate your trouble, both regarding latch and supply, in my opinion. I do not recommend pumping and feeding with a bottle--that will feel labor-intensive and discourage simple breastfeeding even after you get over this speedbump. I also do not recommend using a nipple shield at this stage of the game--and brestfeeding is a confidence game. You're still very much in the range of normal for getting settled in to nursing this baby; things will start to regulate a bit better very soon. 5 to 6 really wet diapers every 24 hours is a good sign, along with at least 3 to 4 yellow stools the size of a U.S quarter or larger. If stools are greenish, this suggests Baby is receiving more foremilk (high protein but not high fat) than hindmilk; consider keeping Baby at just one breast per feeding. If you received IV fluids during labor, it's possible your baby's birth weight was inflated by the extra fluids and her weight gain would seem to start off slowly--keep an eye on it, but if all other signs of good health are present, you probably don't need to worry too much *just* yet.
Every day your lactation hormones are adjusting a bit more and your baby is growing and getting better able to handle your let-down, which should ease up within a couple weeks. Hang in there, mama! I'm thinking of you so keep us posted--I hope this is a bit useful. Best wishes!