I believe that paradise (aka the Spirit World) is currently on the earth in another dimension (as you stated) and that at the resurrection the earth will "be renewed and receive its paradisiacal glory."
Here is some information from the LDS basics section on About.com:
[font=verdana, geneva, helvetica]Where is the Spirit World?
Brigham Young answered this question simply when he said: "Where is the spirit world? It is right here." [/font]
[font=verdana, geneva, helvetica]Prophets have told us that the spirit world is not far from us and if we could rend the veil we would see the spirits of righteous people who have gone on before us. They know our thoughts, feelings and emotions. Speaking of these spirits, President Joseph F. Smith said: "They are as deeply interested in our welfare today, if not with greater capacity, with far more interest behind the veil, than they were in the flesh."
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[font=verdana, geneva, helvetica]http://lds.about.com/library/weekly/previousyears/aa011201a.htm
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Joseph Smith said, "[/font]"The spirits of the just are exalted to a greater and more glorious work; hence they are blessed in their departure to the world of spirits. Enveloped in flaming fire, they are not far from us, and know and understand our thoughts, feelings, and motions, and are often pained therewith"
Parley P. Pratt (not a prophet, but a well-known early leader of the LDS church) wrote that the spirit world "is here on the very planet where we were born." Continuing, he explained: "The earth and all other planets of a like sphere, have their inward or spiritual spheres, as well as their outward, or temporal. The one is peopled by temporal tabernacles, and the other by spirits. A veil is drawn between the one sphere and the other, whereby all the objects in the spiritual sphere are rendered invisible to those in the temporal."