I am wondering if there's a verb that means "pour water" such that I don't have to mention "water" and can just mention the thing I am pouring water onto as a direct object of the verb.
I am __ the flowers.
Actually, using water seems most natural to me. Were you aware of this usage? Or, is there a reason you are rejecting this?
water
verb to pour water on to plants or the soil that they are growing in:
• I've asked my neighbor to water the plants while I'm away.
(Cambridge Dictionary)
Example:
I am watering the plants.
Irrigating is an alternative - generally used for gardens, crops, sports fields, golf courses etc rather than a few flowers.
To water is generally taken to mean by hand, either with a hose or watering can.
To irrigate is generally understood to mean using an irrigation system of pipes, sprays, drips and the like.
https://dictionary.cambridge.org/dictionary/english/irrigate