Toilets of that age weren't always the greatest. But, to make sure they are doing the best they can, you should check a few things.
1. Check the water level in the tank...it should either be at the mark, or if there's no mark, about 1/2" or so below the overflow tube...adjust if required.
2. Note the water level in the toilet bowl...take a bucket of water and slowly pour some in and stop when it appears it isn't going up any more. See if it returns to the original level. If it does not, you may need a new fill valve. A toilet bowl that doesn't start out full, will use some of the water to first refill it, and often, take a second flush to get it to clear...the first refills it properly, the second, gets the job done.
3. Try holding the handle down and see if emptying the whole tank works any better. The flapper valves need to release the specified water, and not all aftermarket ones, even when they fit, are the right ones and close early.
4. Depending on your specific situation, if you used a wax ring with a funnel in it, that in itself may be part of the problem. Many times they work, but there are situations where rather than help (there's debate on if they help at all!), they can be the cause of the clogging...if they won't fit into the toilet flange without getting crimped, you need to seal it without one.
If all of those fail, then it may be new toilet time. If you search around here, more people here (myself included) have and like Toto brand...of the four toilets I've replaced for me and family, we have two Drakes, a Vespin, and a Carlyle...over a number of years, they have been trouble-free, which is most people's situation.