I've done something similar, except for amount of construction loan. I already had a builder and we had an estimated construction cost, which is what we asked for from the bank. I did not rely on the appraisal to tell me how much I needed, but an appraisal was needed for the loan.
I took out a loan for the land. it required a down payment (recovered as equity in construction loan) and closing costs (not recovered). I then did a separate one time close construction loan. equity in the land (initial downpayment) was taken into account. the construction loan included paying off the loan for the land as part of the first draw. it did require a second closing with associated closing costs. at any one time, I only had one loan associated with the build: land, then construction, then final mortgage for completed house with land. if I recall correctly, some construction loans will include some sort of process to ask for more money if you go over what you ask for, but not by too much.
couple stressful points about the process. as stated, not being able to lock a rate on your final mortgage until towards the end of construction. also, the banks let you draw down on the construction loan based on work that has been completed. this can become a problem if you need a bunch of cash to secure a sub or purchase some big ticket items or materials and cant get them on credit. I've heard of complex builds where the owner is trying to put together tens to even a
hundreds of thousands of dollars so they can complete the next phase and then submit to the bank for the next draw (and then you can repay yourself the cash). a good builder will know how to manage the draws to minimize this risk. lastly, going over budget is always possible.