I'm 66, single and sole proprietor of 2 unrelated US businesses. Can the loss from one offset the profit of the other one on my 1040?
Short answer: Yes.
For Federal income tax purposes, you are taxed on your total income, adding up positives and negatives. If business A made, say, $100,000 while business B lost $20,000, then your total income is $80,000, and that's what you'll be taxed on.
As @littleadv says, of course any business losses you claim must qualify as business losses under IRS rules. And yes, there are special rules about losses that the IRS considers "passive".
If you have wage income in addition to business income, business losses don't offset wage income for social security and medicare tax purposes. You can't get a refund of the social security tax deducted from your paycheck.
I don't know if this is relevant to you, but: If you have businesses in different states, each is taxed by that state. For example I have two tiny side businesses, one in Michigan and one in Ohio. Last year the Michigan business made money while the Ohio business lost money. So my federal income was Michigan minus Ohio. My Ohio income was negative so I owed no Ohio income tax. But I couldn't subtract my Ohio losses from my Michigan income for Michigan income tax purposes. Thus, having, say, $10,000 income in Michigan and $10,000 in Ohio would result in lower taxes than $30,000 income in Michigan and a $10,000 loss in Ohio, even though the total income in both cases is the same. And this would be true even if the tax rates in both states were identical.
As long as the losing business is not considered "passive activity" or "hobby", then yes.
Passive Activity is an activity where you do not have to actively do anything to generate income. For example - royalties or rentals.
Hobby is an activity that doesn't generate profit. Generally, if your business doesn't consistently generate profit (the IRS looks at 3 out of the last 5 years), it may be characterized as hobby. For hobby, loss deduction is limited by the hobby income and the 2% AGI threshold.