Players out:
RVP
Falcao
Nani - wasn't involved last year
*Valdes - barely involved last year
*Di Maria
James - barely involved last year (I think one game only)
*Rafael - barely involved last year
Players in:
Darmian
Schneiderlin
Schweinsteiger
Depay
Chicha - coming back from loan
* As it stands at the moment I'm assuming Di Maria, Valdes and Rafael are gone, but DDG, Lindegaard, Evans and Chicha are staying. I'm also assuming that youngsters involved on the tour (Januzaj, Lingard, Pereira, Blackett, Johnstone and McNair) will be staying and not loaned out.
If the above assumptions are right, I think we are in a much better shape even without any more signings. Here is my reasoning:
GK (same as before) - lost Valdes, but we didn't really need him if DDG was staying. Also Johnstone looks good.
RB (big improvement) - lost Rafael, but his injuries prevented him to be really involved in the first team. Darmian looks like the best RB we've had for years. Also Valencia would have gained an extra year of experience as a RB.
LB (slight improvement) - having Shaw fit feels like a new signing. In Blind, Rojo and Blackett we have enough competent backup.
CB (slight improvement) - having solid FBs and DM, plus the super GK would protect our CBs more this season. McNair has also grown in experience and Rojo should have improved his English, plus Blind looks like he might be a decent cover as well. I think even without reinforcement we will be more solid in the back this season.
DM (big improvement) - Schneiderlin was brought as Carrick's replacement I guess, plus Schweinsteiger and Blind can play there too and we still have Carrick

. I would even say that we are the best in the league in this position - only Chelsea with Matic have a player who is arguably better, but they don't have so much depth.
CM (improvement) - Di Maria is the only loss, who could play there, but he barely did for us. Schweinsteiger is a great addition, Schneiderlin could probably play there too, plus Fellaini and Herrera are very solid and maybe Blind could play there as well.
LW/RW (same as before) - lost Di Maria and Nani, added Depay - theoretically we should be worse off after losing two players, especially considering one of them was world class, but first Nani wasn't involved at all last year and second the loss of Di Maria is not as bad as it seems. Yes, he is world class player, but he didn't fit in our team - he had a great start, when he was unplayable for a month and managed be productive with assists even when he wasn't playing well in the second half of the season, however he was frustrating a lot of our fans and even lost his place to Young. On the other hand Depay is very exiting addition and seems like he will fit us like a glove. We also have Young, Januzaj, Wilson, Mata, Pereira and Lingard, who can play on the wing.
CF (slight weakening) - lost RVP and Falcao, added Chicha. This is the only position, where we have weakened, but not as much as it seams at a first sight. First Falcao has barely contributed anything - his loss is not going to be noticed. Second RVP although definitely a big loss, wasn't as influential last year as in previous campaigns. To balance it out we only added Chicha, who many feel has regressed a lot in the past 2 years, but I think he would make a good backup to Rooney. He had very limited opportunities at Real Madrid, but he did very well when he played. On top of that we have Wilson, Januzaj and Depay - there is very good potential there, but the most important reason why we haven't weakened that much is because it looks like LVG will play with one striker this year in a 4-3-3 or 4-2-3-1 formation as opposed to last year, when he played with two upfront. So as a whole we are not that bad in this position as some make it out to be. Now, there is question if Rooney is the same player as before, but this a whole different discussion.
So, I think we have improved or stayed the same in all positions, except CF and things are not that bad there either. Add to that that we have had a year with LVG's methods and everyone knows what is expected from them, plus some very exciting youngsters, who look to have matured a lot and I think I am right to say that we are in a much better shape currently. Of course, things could change a lot if DDG leaves, we let go of Chicha and sell/loan some of the youngsters mentioned above.
Edit: Forgot we also lost Cleverley and Anderson, but this changes nothing, cause they weren't involved enough.