This is a problem of your own making...the good thing is you can change it.
First kick that guilt out the window!! they may miss you if you go out but think of it as you are gently preparing them for their future in a gradual way, this is as equally an important part of parenting as playing or reading or any other aspect.
Stop feeding your 2 year old to sleep there really is no need.
Get a childgate or two (l assume both dcs are quite young) in the doorway of the room their toys are in.
We had a small room off the kitchen when mine were small, which became the playroom, toys were kept in there and l had a childgate enabling me to cook, iron etc without mine (16 months apart) under my feet.
Establish a routine of mummy play time and children play time . where they are told 'it is time for mummy to cook dinner so it is now children play time/TV time' in the play area. If you are in another room use a monitor to watch.
You have to break this clingy habit as it will be a bigger problem later on.
I had 2 very close in age, a DH who mostly worked away and no other support...this is now l created a ( largely) happy peaceful home..
Routine that suited us, me and Dc
Eg up at same time each day 7 am, feed and change youngest, put him in play area with toys.
Have breakfast with DD (2) at table then bring both to bathroom with portable playpen, they stayed in playpen whilst l showered, often l would shower them one by one too, other Dc in playpen on towels.
They stayed in playpen whilst l blowdried my hair and got dressed. Then got them dressed.
Went out, back for lunch ( DS in high chair, dd sat in booster chair)
After, announced 'children nap time' put them in seperate dark bedrooms for naps, cleared up prepped dinner, put washing on etc.
once they woke would play with each for a while individually and together..'mummy playtime',........... they soon understood that 'children playtime ' meant I was doing something else.
Bedtime routine was pretty rigid, as by then l wanted to have some adult time.
Tea at 5.30 , 'mummy playtime' reading and cuddles, DS put in bed for 6.30pm and DD in bed for 7pm and left...
I always had my evenings free by 7.30 until they were 7and 8 respectively. Even now at 10 and 11 they go straight to bed at 8 and 8.30pm with no fuss.
I can imagine how frustrated you are and probably bored, so start establishing boundaries.
Make reward charts that teach them if they DONT shout for you for 2 days, they get a special treat or outing. If they insist they lose a priviledge.
Make the rules very clear, and stick to them, introduce a new rule once you have successfully nailled the first, ie if shouting for you is the main issue then start there. Then add new rules until you have the home life you want..it is perfectly doable.
I used to have colourful star charts in the playroom and refer to them in a positive way each day, discipine does not have to involve stress and shouting, just a quiet determination from you!
Once you have sorted your home life then you can focus on your personal life. Thats the easy bit!
Good luck op!