How do I do it -
Emotionally - I cannot go to a shelter unless I have a very specific job and guidelines because it is too difficult for me otherwise. I give a lot of credit to people who can work and volunteer in shelters.
For fostering, I get very attached too, but know that my job is to find those dogs a better life and home. And will wait patiently until that happens.
For involvement I like to do a lot of things - I like fostering, but know that you can be a good volunteer in other ways without fostering. I do home checks. I do pictures. I do fundraising and try to network. I "coordinate" events. I will also go to events, bring rain, and pretend to be friendly.

I will do online research for things as requested - health issues, best practices, food, whatever. I try to come up with flyers and info stuff. And I did the HW blog, plus have been bad at updating but do the other general blog. Now that I am going to take a little fostering break, I offered to be a foster mentor. There are lots of other things that people do too! Newsletters, reference checking, transports, etc.
I agree with Sarah on the whole
One important thing is to realize that while the skills developed with one's own dogs are helpful they do not prepare all the way for handling many shelter dogs. There is a lot more to learn when working with shelter dogs. It's a great opportunity for handlers with an open mind to both help the animals and expand their own skills.
I met a little rescue girly this weekend who did the blink and cringe thing, waiting to be hit or yelled at but who followed me around, wanting attention - but also afraid of that attention. Luckily she's in a foster home so no need for my help but I just let her do her thing and eventually I was able to pet her.

Thank you for wanting to volunteer! If you go in with an open mind (and I am sure you will!) to what the dogs will teach you, your mind will be B L O W N!