I am not saying that having someone else making contact on your behalf is the best option but rather one of your options. There is no right or wrong way however it is important to consider all the options. For example, a letter allows the found natural mother to adjust to the idea before it is necessary for them to respond.

Even if your natural mother searched for you unsuccessfully the first contact will be a shock and they will need time to work through their feelings.

There are many natural mothers who have never told anyone that they had relinquished a child to adoption. For those natural mothers of the closed adoption era it was a deep dark secret and many were made to feel ashamed of the situation. Having a child out of wedlock was considered by society as shameful. As the years went on, society has changed and this may have made the natural mother feel worse. So, the natural mother may feel very torn between wanting to meet you and know that you are all right and terrified to tell anyone about you.

You may consider sending a picture in that first letter but you need to consider that may be a threat to the person you are contacting. It may be better to mention that you will send a picture soon so that the person receiving the letter can see how much you look like your natural parents.

If you choose to write a letter be sure to include your complete name, address, and phone number. You may also want to consider including your email address as this can be a convenient way to get contact going.
Another option to making contact is by making a phone call. If you decide on this option, it is best to write down just what you want to say before you place the call. It is important to be sure you are speaking with the right person. If the person you are calling is not there try to find out when would be the best time to call. If you are asked who is calling it is okay to say that you are an old friend of hers and leave your name and phone number.