Hi KeeneT,
Sounds like you may have two alternatives.
1. A fulfillment service such as Cafepress, Zazzle, etc. However, as you say, you do not have any customer control.
2. Use an independent company who will take the orders then print, pack and ship directly to your customer.
Using alternative #2, you can either:
A. Setup your own website where you will get the orders, then forward them to your printer for processing or;
B. You can have the printer create the website, take orders directly and process them from there.
Using alternative #2A, you have full control over your customer input, information ,etc... You just have to do the processing of information (sizes, colors, styles, etc..) and send it to your service.
Using alternative 2B, your printer would have full control and be responsible for filling all orders. He would also notify you as to all orders, processing, etc...
Reputable printers would not use your customer contact information to try to sell their services to your customers as they are only interested in printing. Besides, it makes bad business.
Generally speaking, these printers also have a base price that is lower than the larger companies.
To my knowledge, there are few printers who do offer this type of service and if there were any here, or in any other group, rules dictate that they cannot post as such in any posts.
You might try a search in google using print on demand, POD and other such search terms to see what comes up.
Usually, POD printers offer the bare items such as t-shirts, sweats, decals, signs, magnets, etc ... They do not offer a wide variety of products such as Cafepress due to the very small number of people who do use their service and the high cost of maintaining a large inventory.
Hope this helps.
Fred