If you and your spouse file a joint return for the year of sale, you can exclude gain if either spouse meets the ownership test - see IRS publication 523, Selling Your Home - http://www.irs.gov/pub/irs-pdf/p523.pdf page 15
So if one spouse owned the house at least two years and you are filing joint return another spouse for purpose of $500,000 gain exclusion is considered as satisfied the test. Each spouse still need to life in the house at least two years.
The property that you will purchase will have some property tax attached based on its value - that is a taxes you generally should pay.
But as you are above 55 and own the house - you may transfer your current property tax amount to new property.
If your current property tax bill is less than a new bill will be - you may take advantage of proposition 13.
If a new bill will be less than your current property tax liability - you should pay a new bill and the proposition 13 would not apply.
Proposition 13 applies if an old and a new property are in the same country. As you will move to different county - the proposition 90 might apply if Santa Clara county board adopted it allowing transfers from other counties.
Both Alameda and Santa Clara are currently participating counties in this program. - http://www.sccgov.org/portal/site/asr/agencyarticle?path=%252Fv7%252FAssessor%252C%2520Office%2520of%2520the%2520%2528ELO%2529&contentId=0f70bb3166b34010VgnVCMP2200049dc4a92____
Santa Clara county office of the Assessor - http://www.sccgov.org/portal/site/asr/
Form to use - http://www.sccgov.org/SCC/docs/Assessor,%20Office%20of%20the%20(ELO)/attachments/500450BOE60AHcurrent.pdf
Contacts - Assessor Real Property - General Questions, Property valuations and Propositions 3, 8, 60, 90
70 West Hedding St., East Wing
San Jose, CA 95110
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