I see the resistor in the AC side. If I replace it with a smaller one, would it give more current to DC side?
No. That is most likely part of the circuit that limits the current on the input side. The limit is there because the transformer has been designed down to a price and will saturate only just above whatever input current is needed to get 1 A out. So if you increase the current, you will just get the transformer saturating, and it won't be transferring any more power, but everything will get a lot hotter.
That is a cheap, mass produced, switch-mode power supply. All the components are chosen to be as cheap as possible, so they won't have any spare capacity if it costs any money at all. In particular, the transformer will be designed for that application specifically.
Switch-mode power supply design is very complicated. It too hard for most of the contributors to this forum, including those who have years of electronics experience. A change from 1 A to 2 A would certainly need the transformer and the capacitors changing, but the transformer has been custom-made, so there is no simple alternative that would work or would be obtainable.
The alternative of buying the right sort of power supply is so much easier, cheaper, more reliable and safer.
**broken link removed**
$2.05 including shipping from China.
Newark have these:-
https://www.newark.com/pro-elec/28-19338/adaptor-ac-dc-5-1v-2-5a/dp/15AC7490 for a bit more money but you can get it sooner. (They have over 42,000 of that particular one in stock, so they must sell lots)