Should I adjust rear derailleur as my chain seems too close to each other
The chain is too long. Presumably you did not cut it down to size it when you installed it.

All chains are sold with excess length and must be sized to the bike. The length required depends on the length of the chainstays and size of the largest chainring and sprocket.

The easiest way to size the chain is making it the same length as the old chain. You can also find the required length by putting the chain on the bike on the largest chainring and sprocket (and not through the derailleur). This Park Tool video shows how it's done.



Update after getting info that the new chain was cut to length of old chain:

Don't just shorten the chain and see what happens, you need to follow the sizing procedure to make sure the length is right. If the chain is too short for the large chainring/largest sprocket bad things will happen if you attempt to change into that combination.

Before you do that, check that your derailleur has enough total capacity for the chainrings and sprockets you have. Your derailleur looks like a short cage version and may not be able to take up the chain slack necessary.

Total capacity needed is:

(difference in teeth between largest/smallest chainrings) + (difference in teeth between largest/smallest sprockets)

If you have a Shimano derailleur you can look up the specs here. Note there is a link to archive spec docs if you have an older model. If you don't know what model you have or can't figure it out post a pic of the derailleur in your question, and tell us road or MTB and how many cassette sprockets you have and someone will likely identify it for you.