It depends on the language of the first easement.
As a general proposition, an easement does not grant "exclusive use".
The issue would likely be whether the second easement unreasonably interferes with the use of the first easement holder.
I am not a TX attorney, laws vary from state to state, therefore you should always consult a local attorney.
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You need to take the easement and the proposed easement to a real estate attorney. Without knowing the exact terms, it is very difficult to give you further information. The other attorney is correct, the easement could not be exclusive. However, if it does say that, then there is a question of whether it was a conveyance of the fee title (right-of-way deed), meaning, the grantee of that easement owns the easement tract outright. But, this is unlikely.
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It would be important for an attorney to review the actual language in the express easement. It would be very unusual for a grantor to grant an easement for the exclusive benefit of the grantee, as the easement would be on the land of the grantor.
Absent language granting you an exclusive right of use, more likely than not, the grantor would be able to allow others to use the easement as well.