This is something you will have to work out with the board of directors.  Even if you are the sole member of the organization, you probably have a board that governs the organization and its operations.  It is hard to assure the use of funds after you retire or after your death, no matter what type of control you have now, but you could do a couple of things to make it more likely that the assets will continue to be used for animal rescue.

You can assure that the dissolution clause of your articles of incorporation provides that the net remaining assets will be used for animal rescue purposes upon dissolution and not for other charitable purposes.  This might be changed after your death, but you could provide that it can’t be changed for a certain period of time, or without some super-super majority (such as 90%) of the board voting to change it.  You would be dependent upon some friendly directors or the state attorney general to enforce this after your death, but it is likely to be followed at least for a reasonable period of time.

If you think the organization will continue the work after you leave, you could include similar language in the purpose clause of the governing instruments to increase the likelihood that it will continue your work.  There may be people willing to continue the purpose for a long time.

If you don’t think the work will continue within your organization if you leave, you could make arrangements now to assure that the net remaining assets will go to a specific organization doing the same work in your community.  Or you could agree to a merger now, with the other organization as the surviving entity. 

It is very hard to assure the use of funds for a long period after your death, but these are a few of things you might do now to make it more difficult to change.  Whatever you do, unless you are the sole member, officer and director, you can’t do it alone.