OP, thank you for considering volunteering as a rescue helper. 
Firstly, I'm so glad someone has already mentioned the wonderful, brilliant Cinnamon Trust - not only are you helping a dog but you are helping an elderly or poorly owner. This really is a fab charity. Once you are "on the books" you will be paired with a dog (or dogs!) that you can walk regularly. You also get to know the owner. Well worth a look.
Also, there are so many ways that you can help your local rescue that don't necessarily involve the bits that are heart breaking. Most charities need help with things like :- store collections, meet and greets (you'll probably be given a foster dog to take along to meet some potential owners), helping out at events - things like running a stall, or ring stewarding at a charity event. How about doggy baking? Crafting to make fund raising items? Website skills? These are vital, and most rescue websites are embarrassingly bad considering they are the first "shop window" for potential new adopters.
If you can drive, most rescues do a lot of shuttling as dogs need to be moved from pounds into foster or kennel space, and then on to new homes. Transport volunteers are always needed and you get your mileage back.
Many rescues these days have forums where volunteers can be paired up with what needs doing - also many have FB pages where they will also post for help. Once you start helping, you'll be amazed at the ways you can be of help.
Rescue volunteering is incredibly worthwhile, interesting and fun. You'll meet some gorgeous dogs (well, they all are!), some equally lovely people and a few nutters, and you will be amazed that you ever had any spare time after a while. Just to give a flavour - here are just some of the varied things i've done this year
Assisted new adopter with behavioural support
Done an application to the Charity Commission for charitable status
Helped create a website (which then got hacked by Syrian bandits 
Wrote a leaflet
Helped to organise a charitable dog show
Picked up rosettes
Organised judges
Sourced printed programmes, designed and printed posters
Got thank you bottles for judges
Judged at a charity dog show
Fostered dogs
Planned some events for next year
Helping to organise some new ways of giving (by text) for small local charity
Made Christmas cards for charity shop
Transport volunteer
Wrote and researched submissions on puppy farming to Welsh Govt
List could go on but I hope this gives a flavour of the sheer variety!
Go for it!! 