Learn English – “After given” or “after being given”

"Keep score" is a (slightly) idiomatic expression that means "keeping track of the score". It is not ungrammatical to use this phrase on a T-shirt (or anywhere else). If the T-shirt had the text "keep a score", it might actually give the impression that the shirt was not made by a native English speaker.

(This is speaking as someone from the US; I don't know for certain if the expression might be different in the UK.)

The conjunction and introduces a new independent clause, opening with the phrase in doing so and with we as its subject. However the subject we is not the same as the implied subject of the doing, resulting in a classic example of a dangling participle.

The OP's sentence is not as egregious as the following similar constructions found in a quick internet search, but a careful writer would probably want to rewrite it as the OP suggests:

Travelling to Finland, the weather got colder and colder.

Rushing to the catch the bus, Bob's wallet fell out of his pocket.

Walking back home yesterday, a tree nearly fell on my head.

Flitting gaily from flower to flower, the football player watched the bee.