You need to seek care from a reproductive surgeon or reproductive endocrinologist to determine the best option for you. Your dr. Will need to know the specific technique used to cut your fallopian tubes (ideally they would like to review the operation report), your general health and some reproductive fertility
at 44, your success rate is extremely low - I am very sorry. The natural pregnancy rate in the us is about 0.5-1/1000 women. Ivf might be 1-2% (and only if your fertility tests are normal). Tubal reversal is likely a very poor option for you because it takes 12-18 months to achieve maximum pregnancy rates and you simply do have that much time. I am sorry that the stats for women your age are really poor. See link (http://www.Cdc.Gov/art/artreports.Htm).
sterilization reversal surgery can cost almost as much as in ivf cycle because the surgery requires anywhere from 2-4 hours to perform in the operating room is very expensive part of the hospital. The cost is from about $8, 000 - $15, 000. Ivf cycles often come with a refund guarantee but surgery never does - and this may make ivf cheaper.
Now assume all options on the table
sterilization reversal is one of the few reproductive surgeries that offers a high success rate. The surgery requires an operative microscope in order to remove the scar tissue around the ends of the tube that was cut and to sew them together. Not every surgeon is trained to perform microsurgery therefore you should seek out a surgeon with experience.
Ivf does not require that your fallopian tubes be open. Ivf is a good option for situations in which the sterilization surgery removed a large amount of fallopian tube, the woman's age is older than 35-38 years or or there is also a male factor involved such that the sperm
has a low potential for fertilizing the egg. You did not state in your question whether this is the same partner that created the first children
or whether you have a new partner and he may or may not be fertile.
In order to choose between these 2 treatment options, we need to determine that there would be sufficient tube length at the end of the sterilization reversal surgery to yield a high probability of success compared to ivf, that your fertility potential is not compromised suggesting a small window of opportunity to achieve pregnancy, that your husband has normally functioning sperm and that you are willing to invest at least one year after surgery to achieve pregnancy. Additionally, if you're a good surgical candidate and you want to have multiple children then surgery may be a better option because we cannot always guarantee that you will have extra embryos frozen after ivf. Ivf is so success these days that there is a trend away from surgery.
We often cannot reliably determine whether surgery or ivf would be the best option. We also need to look at the costs of both surgery and ivf and what insurance
will pay plus whether you can have a refund guarantee for ivf.
In conclusion, you may have 2 options for achieving another child. I predict that when you meet with your physician, that the answer to some of these questions will become more obvious. I also think you will be steered to ivf if you are a candidate because it works faster. Good luck.