Many doctors are not comfortable inserting an IUD. If not done properly, there can be complications, though an experienced doctor should have no problem. If your past IUD was removed because of an infection or other problem, then your doctor my not want to expose you to the risk of that problem happening again. If you really want to know what your doctor was thinking, you should ask her.
Incontinence associated with coughing or sneezing is most often due to pelvic changes that occur sometimes years after childbirth.
IUDs are available. Modern IUDs can be left in place for many years and provide continuous contraception without having to remember to refill a prescription or to take a pill. Some IUDs have hormones embedded in them, others have copper wire that provides some added measure of protection. All are somewhat costly, but given the years of protection are cheaper on a per year basis. On the other end of the spectrum, you could be fitted for a diaphragm which together with spermicidal jelly provides fairly effective relief for a married couple. You should speak with your doctor who can give you more information.