You sound like a very nice and practical person to be friends with!
Saying no could undermine the friendship. I also ask them for important favors, though never at the last minute. If I say no, they might do the same. If I make excuses, they will, too. This has happened before.
Then don't make excuses or give them a reason to say no or excuses.
We indians are taught from a young age that relationships have to be prioritised and nurtured in our culture, especially by actions like being there for someone when they need us. But obviously, this can sometimes be very tiring and exhausting. Often, we forget to nurture our self - whether it be through a special interest or a hobby or our career etc. - and prioritise which relationship we want to be in more. All that too matters very much in life.
Don't look at it from an "either this or that" perspective - you saying no once or twice or many times, doesn't mean that your relationship has to end or that it (ethically) frees the other person from their obligation to help you (despite owing you for past favours).
You just have to learn the appropriate skills to deal with their tantrum, anger or hurt feelings when you say no to them. After all, ups and downs are normal in a relationship.
Let me give you an example - assume that you are really busy with an important work and one of your relative asks you to help them with something, without warning. You want to really prioritise your own work because it is important. You tell them that their timing is very bad and you are swamped with a lot of work that you cannot put off any longer. When they insist emotionally, you tell them clearly that this time you really cannot help them at this moment but are willing to help them tomorrow or next week. You do not budge, and keep saying no to their pleadings, and they finally get the message.
What happens next? They may understand. All, however, will get upset to some degree.
But that's ok. All of us know that sometimes life's shitty - either because of our own ineptitude or due to the actions of others or just due to uncontrollable circumstances. It's also natural to get upset when someone says no, even if the 'no' is justified. Some may wrongly just blame you for not being a good relative or friend.
Either way, you shouldn't worry about it because you know that you acted with honest intentions, and not with any malice to end your relationship with them. Sometime we do have to put our own need first (once in a while), especially when it is the other person being inconsiderate.
Now, if you never meant to end your relationship, what's next? Well, what do you do when someone you know is upset? You try to cheer them up. And that's what you should do, as soon as possible, after you are done attending to your own work / life. When you are free, follow up with them on what happened. Find out if they still need help and offer to do so. If they are hurt and standoffish, or angry and rant at you for not helping them when they needed, listen to them and assuage their hurt feelings. Be nice. If necessary, scold them too. If they accuse you of not being there, remind them of all the times you did help them, and tell them it hurts when you accuse them of not being helpful because you couldn't. Don't be afraid to share your feelings appropriately and honestly.
My point is, you don't have to cut off the relationship or allow it to die just because someone is upset at you (or is being immature - you be the adult, in that case).
Depending on how much you like them, if they are still hurt and angry, do something nice for them - like giving them a small gift or inviting them home for tea or a meal. Helping someone when they need it, and ask for it, obviously does matter in any relationship. But you don't always have to say yes (when you can't) to convey that you value the relationship. Sometimes it means as much to just being in touch consistently with them to share their happiness or sorrow or daily worries. Or even doing a kind gesture, like following up in person to find out how they are or inviting them over when they are emotional or avoiding you - to just remind them that they are not forgotten and are important in your life too.
Trust me, this is a more balanced and satisfying way to have a relationship. Just be true to yourself and share your feelings honestly with others to the best of your ability - help those you can depending on how badly they need it, but don't be afraid to prioritise your own needs once in a while. And just learn to deal with upset and emotional people with kindness (and a little tough love)
Similarly, do not be afraid to ask for help when you need it, from your friends or family as long as you too understand they have a right to say no, despite whatever favour you have done for them. Some will help, some may not be able to due to their own circumstances. Those who can't help you today, will help you some other time. And some may simply be selfish and not want to help you - with these, there is nothing wrong in reminding them of the past favours you have done them. If they still don't help you, you can stop prioritising your relationship with them.
But never stop being kind to them. And yourself.
