I would agree with much stated above. The hoe is a legitimate period style but you could still find that style up into the 1970's in hardware stores.
The hatchet… I would say it's possible with the caveat that the war produced an immense need for tools and they were supplied from all kinds of places. Some new, some used, some saved from the scrap pile and some brought from home. The corrosion on that hatchet head makes it difficult to identify but the nail pulling notch is more in line with a carpenters hatchet of the time. The thing is they were being made by a lot of companies and individual blacksmiths to the same basic pattern through the 1960's.
Tools were used until they were no longer useable, then they were typically sold or traded to the local blacksmith to be reshaped into something else.
When it comes to tools, firearms or anything else buy the item, not the story. Could these both be period? Absolutely. Could they be from the depression? Yes as well.
I have a variety of period tools, most of mine are actually 1870's or prior. But many of those tools haven't really changed that much in pattern since. Many make the mistake of believing period tools were crude in comparison to today. That is absolutely not true, especially for blacksmith tools.
There was a blacksmith/farrier in every town and many a farmer also dabbled out of necessity. They were skilled by necessity and competition. Their work was typically as good or better than machine made. It was still a viable career into the late 1950's early 60's.
With the corrosion on both items verifying one way or the other would be quite difficult. But even without it could be difficult.
As an example I have a felling ax that looks to be rather modern. The head is 1840's and the handle is 1970's. There was no malice in it the previous owner was just using it. Another I sold on was from the 1870's and the farm owners had notched the head every time it had been re handled. IIRC there were 8 notches on the head that lined up with the farm account book. Yes, the family had accounted for everything purchased for the farm over 130 years when the family left farming.
In short what is pictured are two period style tools dug at such n such… they could have been dropped and lost in 1863 or 1933.