Corelating between the concept of Universal Hashing wrt Algorithms (hash-table use) & the use of Universal Hashing in a one-time MAC
For simplicity, if I assume a single block of message, then how do I use this universal function to evaluate the hash?

We shift some parameters from being part of the concrete function instance into the key.
For your example of $h_{ab}(d)=ad+b\mod p\mod m$ the following mapping is usual:

$d$ is the input to be hashed
$a$ is the secret key of the instantiation, reusable across multiple invocations
$b$ is the whitening value, it must be freshly random and hidden from adversaries for each invocation
$p,m$ are system parameters that depend on in which field you want to work and into which you have to encode your result, e.g. you could pick $p$ to be a relatively large prime and $m=2^k$ for some $k$.

Alternatively, if you insist on having only two non-parameter inputs, you can also force $a$ to have the same usage constraints as $b$ and declare them both &quot;the key&quot;.