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Assuming that the properties are held in the testator's name and not joint tenancy "with right of survivorship" or some form of tenancy that passes the property automatically upon death, then yes, probate would be necessary. Probate is needed to appoint an executor, even though you were named executor in the will. The naming of an executor is the testator indicating his/her wish of whom he/she wants to serve in this role. It's not an actual appointment, as the named executor can refuse, fail to qualify, predecease the testator, etc... so the court does have discretion to appoint someone else. Once you're appointed as the executor by the court, you can then transfer property, sell it, etc...
The terms of the will seem pretty clear, and unless the children are in agreement as to some different plan (the agreement would need to be in writing and signed by all the children, assuming that they're all adults), you would need to sell the properties and distribute the proceeds (after paying off debts) according to the terms of the will.
That being said, you need to contact an attorney in your area that deals with probate cases. Go to www.lawyers.com or www.legalmatch.com to find an attorney in your area. You should be able to find one that will give you a free initial consultation and better advise you of your rights, any problems with your case, likelihood of success, how courts are treating cases such as yours in your area, and what you should do next.
Hope that clears things up a bit. If you have any other questions, please let me know. If not, and you have not yet, please rate my answer AND press the "submit" button, if applicable. Please note that I don't get any credit for the time and effort that I spent on this answer unless and until you rate it positively (3 or more stars). Thank you, ***** ***** luck to you!