Do you have any idea of how old the tyres are? They look a bit cracked in the last photo and they generally only last about 10 years before deteriorating. It may be worth replacing them if you're not sure as they aren't that expensive.
Weight wise they should be fine (28mm is wide for a road bike) but you may need to run them at a higher pressure (within the limits).
There are plenty of online tyre pressure calculators. I can't share links yet but the first one I found on good thinks that you should be running at 104psi (250lb is 113kg).